# Theory Gamma_Function

theory Gamma_Function
imports Equivalence_Lebesgue_Henstock_Integration Harmonic_Numbers
```(*  Title:    HOL/Analysis/Gamma_Function.thy
Author:   Manuel Eberl, TU München
*)

section ‹The Gamma Function›

theory Gamma_Function
imports
Equivalence_Lebesgue_Henstock_Integration
Summation_Tests
Harmonic_Numbers
"HOL-Library.Nonpos_Ints"
"HOL-Library.Periodic_Fun"
begin

text ‹
Several equivalent definitions of the Gamma function and its
most important properties. Also contains the definition and some properties
of the log-Gamma function and the Digamma function and the other Polygamma functions.

Based on the Gamma function, we also prove the Weierstraß product form of the
sin function and, based on this, the solution of the Basel problem (the
sum over all \<^term>‹1 / (n::nat)^2›.
›

lemma pochhammer_eq_0_imp_nonpos_Int:
"pochhammer (x::'a::field_char_0) n = 0 ⟹ x ∈ ℤ⇩≤⇩0"
by (auto simp: pochhammer_eq_0_iff)

lemma closed_nonpos_Ints [simp]: "closed (ℤ⇩≤⇩0 :: 'a :: real_normed_algebra_1 set)"
proof -
have "ℤ⇩≤⇩0 = (of_int ` {n. n ≤ 0} :: 'a set)"
by (auto elim!: nonpos_Ints_cases intro!: nonpos_Ints_of_int)
also have "closed …" by (rule closed_of_int_image)
finally show ?thesis .
qed

lemma plus_one_in_nonpos_Ints_imp: "z + 1 ∈ ℤ⇩≤⇩0 ⟹ z ∈ ℤ⇩≤⇩0"
using nonpos_Ints_diff_Nats[of "z+1" "1"] by simp_all

lemma of_int_in_nonpos_Ints_iff:
"(of_int n :: 'a :: ring_char_0) ∈ ℤ⇩≤⇩0 ⟷ n ≤ 0"
by (auto simp: nonpos_Ints_def)

lemma one_plus_of_int_in_nonpos_Ints_iff:
"(1 + of_int n :: 'a :: ring_char_0) ∈ ℤ⇩≤⇩0 ⟷ n ≤ -1"
proof -
have "1 + of_int n = (of_int (n + 1) :: 'a)" by simp
also have "… ∈ ℤ⇩≤⇩0 ⟷ n + 1 ≤ 0" by (subst of_int_in_nonpos_Ints_iff) simp_all
also have "… ⟷ n ≤ -1" by presburger
finally show ?thesis .
qed

lemma one_minus_of_nat_in_nonpos_Ints_iff:
"(1 - of_nat n :: 'a :: ring_char_0) ∈ ℤ⇩≤⇩0 ⟷ n > 0"
proof -
have "(1 - of_nat n :: 'a) = of_int (1 - int n)" by simp
also have "… ∈ ℤ⇩≤⇩0 ⟷ n > 0" by (subst of_int_in_nonpos_Ints_iff) presburger
finally show ?thesis .
qed

lemma fraction_not_in_ints:
assumes "¬(n dvd m)" "n ≠ 0"
shows   "of_int m / of_int n ∉ (ℤ :: 'a :: {division_ring,ring_char_0} set)"
proof
assume "of_int m / (of_int n :: 'a) ∈ ℤ"
then obtain k where "of_int m / of_int n = (of_int k :: 'a)" by (elim Ints_cases)
with assms have "of_int m = (of_int (k * n) :: 'a)" by (auto simp add: field_split_simps)
hence "m = k * n" by (subst (asm) of_int_eq_iff)
hence "n dvd m" by simp
with assms(1) show False by contradiction
qed

lemma fraction_not_in_nats:
assumes "¬n dvd m" "n ≠ 0"
shows   "of_int m / of_int n ∉ (ℕ :: 'a :: {division_ring,ring_char_0} set)"
proof
assume "of_int m / of_int n ∈ (ℕ :: 'a set)"
also note Nats_subset_Ints
finally have "of_int m / of_int n ∈ (ℤ :: 'a set)" .
moreover have "of_int m / of_int n ∉ (ℤ :: 'a set)"
using assms by (intro fraction_not_in_ints)
qed

lemma not_in_Ints_imp_not_in_nonpos_Ints: "z ∉ ℤ ⟹ z ∉ ℤ⇩≤⇩0"
by (auto simp: Ints_def nonpos_Ints_def)

lemma double_in_nonpos_Ints_imp:
assumes "2 * (z :: 'a :: field_char_0) ∈ ℤ⇩≤⇩0"
shows   "z ∈ ℤ⇩≤⇩0 ∨ z + 1/2 ∈ ℤ⇩≤⇩0"
proof-
from assms obtain k where k: "2 * z = - of_nat k" by (elim nonpos_Ints_cases')
thus ?thesis by (cases "even k") (auto elim!: evenE oddE simp: field_simps)
qed

lemma sin_series: "(λn. ((-1)^n / fact (2*n+1)) *⇩R z^(2*n+1)) sums sin z"
proof -
from sin_converges[of z] have "(λn. sin_coeff n *⇩R z^n) sums sin z" .
also have "(λn. sin_coeff n *⇩R z^n) sums sin z ⟷
(λn. ((-1)^n / fact (2*n+1)) *⇩R z^(2*n+1)) sums sin z"
by (subst sums_mono_reindex[of "λn. 2*n+1", symmetric])
(auto simp: sin_coeff_def strict_mono_def ac_simps elim!: oddE)
finally show ?thesis .
qed

lemma cos_series: "(λn. ((-1)^n / fact (2*n)) *⇩R z^(2*n)) sums cos z"
proof -
from cos_converges[of z] have "(λn. cos_coeff n *⇩R z^n) sums cos z" .
also have "(λn. cos_coeff n *⇩R z^n) sums cos z ⟷
(λn. ((-1)^n / fact (2*n)) *⇩R z^(2*n)) sums cos z"
by (subst sums_mono_reindex[of "λn. 2*n", symmetric])
(auto simp: cos_coeff_def strict_mono_def ac_simps elim!: evenE)
finally show ?thesis .
qed

lemma sin_z_over_z_series:
fixes z :: "'a :: {real_normed_field,banach}"
assumes "z ≠ 0"
shows   "(λn. (-1)^n / fact (2*n+1) * z^(2*n)) sums (sin z / z)"
proof -
from sin_series[of z] have "(λn. z * ((-1)^n / fact (2*n+1)) * z^(2*n)) sums sin z"
from sums_mult[OF this, of "inverse z"] and assms show ?thesis
qed

lemma sin_z_over_z_series':
fixes z :: "'a :: {real_normed_field,banach}"
assumes "z ≠ 0"
shows   "(λn. sin_coeff (n+1) *⇩R z^n) sums (sin z / z)"
proof -
from sums_split_initial_segment[OF sin_converges[of z], of 1]
have "(λn. z * (sin_coeff (n+1) *⇩R z ^ n)) sums sin z" by simp
from sums_mult[OF this, of "inverse z"] assms show ?thesis by (simp add: field_simps)
qed

lemma has_field_derivative_sin_z_over_z:
fixes A :: "'a :: {real_normed_field,banach} set"
shows "((λz. if z = 0 then 1 else sin z / z) has_field_derivative 0) (at 0 within A)"
(is "(?f has_field_derivative ?f') _")
proof (rule has_field_derivative_at_within)
have "((λz::'a. ∑n. of_real (sin_coeff (n+1)) * z^n)
has_field_derivative (∑n. diffs (λn. of_real (sin_coeff (n+1))) n * 0^n)) (at 0)"
proof (rule termdiffs_strong)
from summable_ignore_initial_segment[OF sums_summable[OF sin_converges[of "1::'a"]], of 1]
show "summable (λn. of_real (sin_coeff (n+1)) * (1::'a)^n)" by (simp add: of_real_def)
qed simp
also have "(λz::'a. ∑n. of_real (sin_coeff (n+1)) * z^n) = ?f"
proof
fix z
show "(∑n. of_real (sin_coeff (n+1)) * z^n)  = ?f z"
by (cases "z = 0") (insert sin_z_over_z_series'[of z],
qed
also have "(∑n. diffs (λn. of_real (sin_coeff (n + 1))) n * (0::'a) ^ n) =
diffs (λn. of_real (sin_coeff (Suc n))) 0" by simp
also have "… = 0" by (simp add: sin_coeff_def diffs_def)
finally show "((λz::'a. if z = 0 then 1 else sin z / z) has_field_derivative 0) (at 0)" .
qed

lemma round_Re_minimises_norm:
"norm ((z::complex) - of_int m) ≥ norm (z - of_int (round (Re z)))"
proof -
let ?n = "round (Re z)"
have "norm (z - of_int ?n) = sqrt ((Re z - of_int ?n)⇧2 + (Im z)⇧2)"
also have "¦Re z - of_int ?n¦ ≤ ¦Re z - of_int m¦" by (rule round_diff_minimal)
hence "sqrt ((Re z - of_int ?n)⇧2 + (Im z)⇧2) ≤ sqrt ((Re z - of_int m)⇧2 + (Im z)⇧2)"
also have "… = norm (z - of_int m)" by (simp add: cmod_def)
finally show ?thesis .
qed

lemma Re_pos_in_ball:
assumes "Re z > 0" "t ∈ ball z (Re z/2)"
shows   "Re t > 0"
proof -
have "Re (z - t) ≤ norm (z - t)" by (rule complex_Re_le_cmod)
also from assms have "… < Re z / 2" by (simp add: dist_complex_def)
finally show "Re t > 0" using assms by simp
qed

lemma no_nonpos_Int_in_ball_complex:
assumes "Re z > 0" "t ∈ ball z (Re z/2)"
shows   "t ∉ ℤ⇩≤⇩0"
using Re_pos_in_ball[OF assms] by (force elim!: nonpos_Ints_cases)

lemma no_nonpos_Int_in_ball:
assumes "t ∈ ball z (dist z (round (Re z)))"
shows   "t ∉ ℤ⇩≤⇩0"
proof
assume "t ∈ ℤ⇩≤⇩0"
then obtain n where "t = of_int n" by (auto elim!: nonpos_Ints_cases)
have "dist z (of_int n) ≤ dist z t + dist t (of_int n)" by (rule dist_triangle)
also from assms have "dist z t < dist z (round (Re z))" by simp
also have "… ≤ dist z (of_int n)"
using round_Re_minimises_norm[of z] by (simp add: dist_complex_def)
finally have "dist t (of_int n) > 0" by simp
with ‹t = of_int n› show False by simp
qed

lemma no_nonpos_Int_in_ball':
assumes "(z :: 'a :: {euclidean_space,real_normed_algebra_1}) ∉ ℤ⇩≤⇩0"
obtains d where "d > 0" "⋀t. t ∈ ball z d ⟹ t ∉ ℤ⇩≤⇩0"
proof (rule that)
from assms show "setdist {z} ℤ⇩≤⇩0 > 0" by (subst setdist_gt_0_compact_closed) auto
next
fix t assume "t ∈ ball z (setdist {z} ℤ⇩≤⇩0)"
thus "t ∉ ℤ⇩≤⇩0" using setdist_le_dist[of z "{z}" t "ℤ⇩≤⇩0"] by force
qed

lemma no_nonpos_Real_in_ball:
assumes z: "z ∉ ℝ⇩≤⇩0" and t: "t ∈ ball z (if Im z = 0 then Re z / 2 else abs (Im z) / 2)"
shows   "t ∉ ℝ⇩≤⇩0"
using z
proof (cases "Im z = 0")
assume A: "Im z = 0"
with z have "Re z > 0" by (force simp add: complex_nonpos_Reals_iff)
with t A Re_pos_in_ball[of z t] show ?thesis by (force simp add: complex_nonpos_Reals_iff)
next
assume A: "Im z ≠ 0"
have "abs (Im z) - abs (Im t) ≤ abs (Im z - Im t)" by linarith
also have "… = abs (Im (z - t))" by simp
also have "… ≤ norm (z - t)" by (rule abs_Im_le_cmod)
also from A t have "… ≤ abs (Im z) / 2" by (simp add: dist_complex_def)
finally have "abs (Im t) > 0" using A by simp
thus ?thesis by (force simp add: complex_nonpos_Reals_iff)
qed

subsection ‹The Euler form and the logarithmic Gamma function›

text ‹
We define the Gamma function by first defining its multiplicative inverse ‹rGamma›.
This is more convenient because ‹rGamma› is entire, which makes proofs of its
properties more convenient because one does not have to watch out for discontinuities.
(e.g. ‹rGamma› fulfils ‹rGamma z = z * rGamma (z + 1)› everywhere, whereas the ‹Γ› function
does not fulfil the analogous equation on the non-positive integers)

We define the ‹Γ› function (resp.\ its reciprocale) in the Euler form. This form has the advantage
that it is a relatively simple limit that converges everywhere. The limit at the poles is 0
(due to division by 0). The functional equation ‹Gamma (z + 1) = z * Gamma z› follows
immediately from the definition.
›

definition✐‹tag important› Gamma_series :: "('a :: {banach,real_normed_field}) ⇒ nat ⇒ 'a" where
"Gamma_series z n = fact n * exp (z * of_real (ln (of_nat n))) / pochhammer z (n+1)"

definition Gamma_series' :: "('a :: {banach,real_normed_field}) ⇒ nat ⇒ 'a" where
"Gamma_series' z n = fact (n - 1) * exp (z * of_real (ln (of_nat n))) / pochhammer z n"

definition rGamma_series :: "('a :: {banach,real_normed_field}) ⇒ nat ⇒ 'a" where
"rGamma_series z n = pochhammer z (n+1) / (fact n * exp (z * of_real (ln (of_nat n))))"

lemma Gamma_series_altdef: "Gamma_series z n = inverse (rGamma_series z n)"
and rGamma_series_altdef: "rGamma_series z n = inverse (Gamma_series z n)"
unfolding Gamma_series_def rGamma_series_def by simp_all

lemma rGamma_series_minus_of_nat:
"eventually (λn. rGamma_series (- of_nat k) n = 0) sequentially"
using eventually_ge_at_top[of k]
by eventually_elim (auto simp: rGamma_series_def pochhammer_of_nat_eq_0_iff)

lemma Gamma_series_minus_of_nat:
"eventually (λn. Gamma_series (- of_nat k) n = 0) sequentially"
using eventually_ge_at_top[of k]
by eventually_elim (auto simp: Gamma_series_def pochhammer_of_nat_eq_0_iff)

lemma Gamma_series'_minus_of_nat:
"eventually (λn. Gamma_series' (- of_nat k) n = 0) sequentially"
using eventually_gt_at_top[of k]
by eventually_elim (auto simp: Gamma_series'_def pochhammer_of_nat_eq_0_iff)

lemma rGamma_series_nonpos_Ints_LIMSEQ: "z ∈ ℤ⇩≤⇩0 ⟹ rGamma_series z ⇢ 0"
by (elim nonpos_Ints_cases', hypsubst, subst tendsto_cong, rule rGamma_series_minus_of_nat, simp)

lemma Gamma_series_nonpos_Ints_LIMSEQ: "z ∈ ℤ⇩≤⇩0 ⟹ Gamma_series z ⇢ 0"
by (elim nonpos_Ints_cases', hypsubst, subst tendsto_cong, rule Gamma_series_minus_of_nat, simp)

lemma Gamma_series'_nonpos_Ints_LIMSEQ: "z ∈ ℤ⇩≤⇩0 ⟹ Gamma_series' z ⇢ 0"
by (elim nonpos_Ints_cases', hypsubst, subst tendsto_cong, rule Gamma_series'_minus_of_nat, simp)

lemma Gamma_series_Gamma_series':
assumes z: "z ∉ ℤ⇩≤⇩0"
shows   "(λn. Gamma_series' z n / Gamma_series z n) ⇢ 1"
proof (rule Lim_transform_eventually)
from eventually_gt_at_top[of "0::nat"]
show "eventually (λn. z / of_nat n + 1 = Gamma_series' z n / Gamma_series z n) sequentially"
proof eventually_elim
fix n :: nat assume n: "n > 0"
from n z have "Gamma_series' z n / Gamma_series z n = (z + of_nat n) / of_nat n"
by (cases n, simp)
(auto simp add: Gamma_series_def Gamma_series'_def pochhammer_rec'
dest: pochhammer_eq_0_imp_nonpos_Int plus_of_nat_eq_0_imp)
also from n have "… = z / of_nat n + 1" by (simp add: field_split_simps)
finally show "z / of_nat n + 1 = Gamma_series' z n / Gamma_series z n" ..
qed
have "(λx. z / of_nat x) ⇢ 0"
by (rule tendsto_norm_zero_cancel)
(insert tendsto_mult[OF tendsto_const[of "norm z"] lim_inverse_n],
from tendsto_add[OF this tendsto_const[of 1]] show "(λn. z / of_nat n + 1) ⇢ 1" by simp
qed

text ‹
We now show that the series that defines the ‹Γ› function in the Euler form converges
and that the function defined by it is continuous on the complex halfspace with positive
real part.

We do this by showing that the logarithm of the Euler series is continuous and converges
locally uniformly, which means that the log-Gamma function defined by its limit is also
continuous.

This will later allow us to lift holomorphicity and continuity from the log-Gamma
function to the inverse of the Gamma function, and from that to the Gamma function itself.
›

definition✐‹tag important› ln_Gamma_series :: "('a :: {banach,real_normed_field,ln}) ⇒ nat ⇒ 'a" where
"ln_Gamma_series z n = z * ln (of_nat n) - ln z - (∑k=1..n. ln (z / of_nat k + 1))"

definition✐‹tag unimportant› ln_Gamma_series' :: "('a :: {banach,real_normed_field,ln}) ⇒ nat ⇒ 'a" where
"ln_Gamma_series' z n =
- euler_mascheroni*z - ln z + (∑k=1..n. z / of_nat n - ln (z / of_nat k + 1))"

definition ln_Gamma :: "('a :: {banach,real_normed_field,ln}) ⇒ 'a" where
"ln_Gamma z = lim (ln_Gamma_series z)"

text ‹
We now show that the log-Gamma series converges locally uniformly for all complex numbers except
the non-positive integers. We do this by proving that the series is locally Cauchy.
›

context
begin

private lemma ln_Gamma_series_complex_converges_aux:
fixes z :: complex and k :: nat
assumes z: "z ≠ 0" and k: "of_nat k ≥ 2*norm z" "k ≥ 2"
shows "norm (z * ln (1 - 1/of_nat k) + ln (z/of_nat k + 1)) ≤ 2*(norm z + norm z^2) / of_nat k^2"
proof -
let ?k = "of_nat k :: complex" and ?z = "norm z"
have "z *ln (1 - 1/?k) + ln (z/?k+1) = z*(ln (1 - 1/?k :: complex) + 1/?k) + (ln (1+z/?k) - z/?k)"
also have "norm ... ≤ ?z * norm (ln (1-1/?k) + 1/?k :: complex) + norm (ln (1+z/?k) - z/?k)"
by (subst norm_mult [symmetric], rule norm_triangle_ineq)
also have "norm (Ln (1 + -1/?k) - (-1/?k)) ≤ (norm (-1/?k))⇧2 / (1 - norm(-1/?k))"
using k by (intro Ln_approx_linear) (simp add: norm_divide)
hence "?z * norm (ln (1-1/?k) + 1/?k) ≤ ?z * ((norm (1/?k))^2 / (1 - norm (1/?k)))"
by (intro mult_left_mono) simp_all
also have "... ≤ (?z * (of_nat k / (of_nat k - 1))) / of_nat k^2" using k
by (simp add: field_simps power2_eq_square norm_divide)
also have "... ≤ (?z * 2) / of_nat k^2" using k
by (intro divide_right_mono mult_left_mono) (simp_all add: field_simps)
also have "norm (ln (1+z/?k) - z/?k) ≤ norm (z/?k)^2 / (1 - norm (z/?k))" using k
by (intro Ln_approx_linear) (simp add: norm_divide)
hence "norm (ln (1+z/?k) - z/?k) ≤ ?z^2 / of_nat k^2 / (1 - ?z / of_nat k)"
also have "... ≤ (?z^2 * (of_nat k / (of_nat k - ?z))) / of_nat k^2" using k
also have "... ≤ (?z^2 * 2) / of_nat k^2" using k
by (intro divide_right_mono mult_left_mono) (simp_all add: field_simps)
finally show ?thesis by (simp only: distrib_left mult.commute)
qed

lemma ln_Gamma_series_complex_converges:
assumes z: "z ∉ ℤ⇩≤⇩0"
assumes d: "d > 0" "⋀n. n ∈ ℤ⇩≤⇩0 ⟹ norm (z - of_int n) > d"
shows "uniformly_convergent_on (ball z d) (λn z. ln_Gamma_series z n :: complex)"
proof (intro Cauchy_uniformly_convergent uniformly_Cauchy_onI')
fix e :: real assume e: "e > 0"
define e'' where "e'' = (SUP t∈ball z d. norm t + norm t^2)"
define e' where "e' = e / (2*e'')"
have "bounded ((λt. norm t + norm t^2) ` cball z d)"
by (intro compact_imp_bounded compact_continuous_image) (auto intro!: continuous_intros)
hence "bounded ((λt. norm t + norm t^2) ` ball z d)" by (rule bounded_subset) auto
hence bdd: "bdd_above ((λt. norm t + norm t^2) ` ball z d)" by (rule bounded_imp_bdd_above)

with z d(1) d(2)[of "-1"] have e''_pos: "e'' > 0" unfolding e''_def
by (subst less_cSUP_iff) (auto intro!: add_pos_nonneg bexI[of _ z])
have e'': "norm t + norm t^2 ≤ e''" if "t ∈ ball z d" for t unfolding e''_def using that
by (rule cSUP_upper[OF _ bdd])
from e z e''_pos have e': "e' > 0" unfolding e'_def

have "summable (λk. inverse ((real_of_nat k)^2))"
by (rule inverse_power_summable) simp
from summable_partial_sum_bound[OF this e'] guess M . note M = this

define N where "N = max 2 (max (nat ⌈2 * (norm z + d)⌉) M)"
{
from d have "⌈2 * (cmod z + d)⌉ ≥ ⌈0::real⌉"
by (intro ceiling_mono mult_nonneg_nonneg add_nonneg_nonneg) simp_all
hence "2 * (norm z + d) ≤ of_nat (nat ⌈2 * (norm z + d)⌉)" unfolding N_def
also have "... ≤ of_nat N" unfolding N_def
by (subst of_nat_le_iff) (rule max.coboundedI2, rule max.cobounded1)
finally have "of_nat N ≥ 2 * (norm z + d)" .
moreover have "N ≥ 2" "N ≥ M" unfolding N_def by simp_all
moreover have "(∑k=m..n. 1/(of_nat k)⇧2) < e'" if "m ≥ N" for m n
using M[OF order.trans[OF ‹N ≥ M› that]] unfolding real_norm_def
by (subst (asm) abs_of_nonneg) (auto intro: sum_nonneg simp: field_split_simps)
moreover note calculation
} note N = this

show "∃M. ∀t∈ball z d. ∀m≥M. ∀n>m. dist (ln_Gamma_series t m) (ln_Gamma_series t n) < e"
unfolding dist_complex_def
proof (intro exI[of _ N] ballI allI impI)
fix t m n assume t: "t ∈ ball z d" and mn: "m ≥ N" "n > m"
from d(2)[of 0] t have "0 < dist z 0 - dist z t" by (simp add: field_simps dist_complex_def)
also have "dist z 0 - dist z t ≤ dist 0 t" using dist_triangle[of 0 z t]
finally have t_nz: "t ≠ 0" by auto

have "norm t ≤ norm z + norm (t - z)" by (rule norm_triangle_sub)
also from t have "norm (t - z) < d" by (simp add: dist_complex_def norm_minus_commute)
also have "2 * (norm z + d) ≤ of_nat N" by (rule N)
also have "N ≤ m" by (rule mn)
finally have norm_t: "2 * norm t < of_nat m" by simp

have "ln_Gamma_series t m - ln_Gamma_series t n =
(-(t * Ln (of_nat n)) - (-(t * Ln (of_nat m)))) +
((∑k=1..n. Ln (t / of_nat k + 1)) - (∑k=1..m. Ln (t / of_nat k + 1)))"
also have "(∑k=1..n. Ln (t / of_nat k + 1)) - (∑k=1..m. Ln (t / of_nat k + 1)) =
(∑k∈{1..n}-{1..m}. Ln (t / of_nat k + 1))" using mn
also from mn have "{1..n}-{1..m} = {Suc m..n}" by fastforce
also have "-(t * Ln (of_nat n)) - (-(t * Ln (of_nat m))) =
(∑k = Suc m..n. t * Ln (of_nat (k - 1)) - t * Ln (of_nat k))" using mn
by (subst sum_telescope'' [symmetric]) simp_all
also have "... = (∑k = Suc m..n. t * Ln (of_nat (k - 1) / of_nat k))" using mn N
by (intro sum_cong_Suc)
(simp_all del: of_nat_Suc add: field_simps Ln_of_nat Ln_of_nat_over_of_nat)
also have "of_nat (k - 1) / of_nat k = 1 - 1 / (of_nat k :: complex)" if "k ∈ {Suc m..n}" for k
using that of_nat_eq_0_iff[of "Suc i" for i] by (cases k) (simp_all add: field_split_simps)
hence "(∑k = Suc m..n. t * Ln (of_nat (k - 1) / of_nat k)) =
(∑k = Suc m..n. t * Ln (1 - 1 / of_nat k))" using mn N
by (intro sum.cong) simp_all
also note sum.distrib [symmetric]
also have "norm (∑k=Suc m..n. t * Ln (1 - 1/of_nat k) + Ln (t/of_nat k + 1)) ≤
(∑k=Suc m..n. 2 * (norm t + (norm t)⇧2) / (real_of_nat k)⇧2)" using t_nz N(2) mn norm_t
by (intro order.trans[OF norm_sum sum_mono[OF ln_Gamma_series_complex_converges_aux]]) simp_all
also have "... ≤ 2 * (norm t + norm t^2) * (∑k=Suc m..n. 1 / (of_nat k)⇧2)"
also have "... < 2 * (norm t + norm t^2) * e'" using mn z t_nz
by (intro mult_strict_left_mono N mult_pos_pos add_pos_pos) simp_all
also from e''_pos have "... = e * ((cmod t + (cmod t)⇧2) / e'')"
by (simp add: e'_def field_simps power2_eq_square)
also from e''[OF t] e''_pos e
have "… ≤ e * 1" by (intro mult_left_mono) (simp_all add: field_simps)
finally show "norm (ln_Gamma_series t m - ln_Gamma_series t n) < e" by simp
qed
qed

end

lemma ln_Gamma_series_complex_converges':
assumes z: "(z :: complex) ∉ ℤ⇩≤⇩0"
shows "∃d>0. uniformly_convergent_on (ball z d) (λn z. ln_Gamma_series z n)"
proof -
define d' where "d' = Re z"
define d where "d = (if d' > 0 then d' / 2 else norm (z - of_int (round d')) / 2)"
have "of_int (round d') ∈ ℤ⇩≤⇩0" if "d' ≤ 0" using that
by (intro nonpos_Ints_of_int) (simp_all add: round_def)
with assms have d_pos: "d > 0" unfolding d_def by (force simp: not_less)

have "d < cmod (z - of_int n)" if "n ∈ ℤ⇩≤⇩0" for n
proof (cases "Re z > 0")
case True
from nonpos_Ints_nonpos[OF that] have n: "n ≤ 0" by simp
from True have "d = Re z/2" by (simp add: d_def d'_def)
also from n True have "… < Re (z - of_int n)" by simp
also have "… ≤ norm (z - of_int n)" by (rule complex_Re_le_cmod)
finally show ?thesis .
next
case False
with assms nonpos_Ints_of_int[of "round (Re z)"]
have "z ≠ of_int (round d')" by (auto simp: not_less)
with False have "d < norm (z - of_int (round d'))" by (simp add: d_def d'_def)
also have "… ≤ norm (z - of_int n)" unfolding d'_def by (rule round_Re_minimises_norm)
finally show ?thesis .
qed
hence conv: "uniformly_convergent_on (ball z d) (λn z. ln_Gamma_series z n)"
by (intro ln_Gamma_series_complex_converges d_pos z) simp_all
from d_pos conv show ?thesis by blast
qed

lemma ln_Gamma_series_complex_converges'': "(z :: complex) ∉ ℤ⇩≤⇩0 ⟹ convergent (ln_Gamma_series z)"
by (drule ln_Gamma_series_complex_converges') (auto intro: uniformly_convergent_imp_convergent)

theorem ln_Gamma_complex_LIMSEQ: "(z :: complex) ∉ ℤ⇩≤⇩0 ⟹ ln_Gamma_series z ⇢ ln_Gamma z"
using ln_Gamma_series_complex_converges'' by (simp add: convergent_LIMSEQ_iff ln_Gamma_def)

lemma exp_ln_Gamma_series_complex:
assumes "n > 0" "z ∉ ℤ⇩≤⇩0"
shows   "exp (ln_Gamma_series z n :: complex) = Gamma_series z n"
proof -
from assms obtain m where m: "n = Suc m" by (cases n) blast
from assms have "z ≠ 0" by (intro notI) auto
with assms have "exp (ln_Gamma_series z n) =
(of_nat n) powr z / (z * (∏k=1..n. exp (Ln (z / of_nat k + 1))))"
unfolding ln_Gamma_series_def powr_def by (simp add: exp_diff exp_sum)
also from assms have "(∏k=1..n. exp (Ln (z / of_nat k + 1))) = (∏k=1..n. z / of_nat k + 1)"
by (intro prod.cong[OF refl], subst exp_Ln) (auto simp: field_simps plus_of_nat_eq_0_imp)
also have "... = (∏k=1..n. z + k) / fact n"
(subst prod_dividef [symmetric], simp_all add: field_simps)
also from m have "z * ... = (∏k=0..n. z + k) / fact n"
by (simp add: prod.atLeast0_atMost_Suc_shift prod.atLeast_Suc_atMost_Suc_shift del: prod.cl_ivl_Suc)
also have "(∏k=0..n. z + k) = pochhammer z (Suc n)"
unfolding pochhammer_prod
also have "of_nat n powr z / (pochhammer z (Suc n) / fact n) = Gamma_series z n"
unfolding Gamma_series_def using assms by (simp add: field_split_simps powr_def)
finally show ?thesis .
qed

lemma ln_Gamma_series'_aux:
assumes "(z::complex) ∉ ℤ⇩≤⇩0"
shows   "(λk. z / of_nat (Suc k) - ln (1 + z / of_nat (Suc k))) sums
(ln_Gamma z + euler_mascheroni * z + ln z)" (is "?f sums ?s")
unfolding sums_def
proof (rule Lim_transform)
show "(λn. ln_Gamma_series z n + of_real (harm n - ln (of_nat n)) * z + ln z) ⇢ ?s"
(is "?g ⇢ _")
by (intro tendsto_intros ln_Gamma_complex_LIMSEQ euler_mascheroni_LIMSEQ_of_real assms)

have A: "eventually (λn. (∑k<n. ?f k) - ?g n = 0) sequentially"
using eventually_gt_at_top[of "0::nat"]
proof eventually_elim
fix n :: nat assume n: "n > 0"
have "(∑k<n. ?f k) = (∑k=1..n. z / of_nat k - ln (1 + z / of_nat k))"
by (subst atLeast0LessThan [symmetric], subst sum.shift_bounds_Suc_ivl [symmetric],
subst atLeastLessThanSuc_atLeastAtMost) simp_all
also have "… = z * of_real (harm n) - (∑k=1..n. ln (1 + z / of_nat k))"
by (simp add: harm_def sum_subtractf sum_distrib_left divide_inverse)
also from n have "… - ?g n = 0"
by (simp add: ln_Gamma_series_def sum_subtractf algebra_simps Ln_of_nat)
finally show "(∑k<n. ?f k) - ?g n = 0" .
qed
show "(λn. (∑k<n. ?f k) - ?g n) ⇢ 0" by (subst tendsto_cong[OF A]) simp_all
qed

lemma uniformly_summable_deriv_ln_Gamma:
assumes z: "(z :: 'a :: {real_normed_field,banach}) ≠ 0" and d: "d > 0" "d ≤ norm z/2"
shows "uniformly_convergent_on (ball z d)
(λk z. ∑i<k. inverse (of_nat (Suc i)) - inverse (z + of_nat (Suc i)))"
(is "uniformly_convergent_on _ (λk z. ∑i<k. ?f i z)")
proof (rule Weierstrass_m_test'_ev)
{
fix t assume t: "t ∈ ball z d"
have "norm z = norm (t + (z - t))" by simp
have "norm (t + (z - t)) ≤ norm t + norm (z - t)" by (rule norm_triangle_ineq)
also from t d have "norm (z - t) < norm z / 2" by (simp add: dist_norm)
finally have A: "norm t > norm z / 2" using z by (simp add: field_simps)

have "norm t = norm (z + (t - z))" by simp
also have "… ≤ norm z + norm (t - z)" by (rule norm_triangle_ineq)
also from t d have "norm (t - z) ≤ norm z / 2" by (simp add: dist_norm norm_minus_commute)
also from z have "… < norm z" by simp
finally have B: "norm t < 2 * norm z" by simp
note A B
} note ball = this

show "eventually (λn. ∀t∈ball z d. norm (?f n t) ≤ 4 * norm z * inverse (of_nat (Suc n)^2)) sequentially"
using eventually_gt_at_top apply eventually_elim
proof safe
fix t :: 'a assume t: "t ∈ ball z d"
from z ball[OF t] have t_nz: "t ≠ 0" by auto
fix n :: nat assume n: "n > nat ⌈4 * norm z⌉"
from ball[OF t] t_nz have "4 * norm z > 2 * norm t" by simp
also from n have "…  < of_nat n" by linarith
finally have n: "of_nat n > 2 * norm t" .
hence "of_nat n > norm t" by simp
hence t': "t ≠ -of_nat (Suc n)" by (intro notI) (simp del: of_nat_Suc)

with t_nz have "?f n t = 1 / (of_nat (Suc n) * (1 + of_nat (Suc n)/t))"
also have "norm … = inverse (of_nat (Suc n)) * inverse (norm (of_nat (Suc n)/t + 1))"
also {
from z t_nz ball[OF t] have "of_nat (Suc n) / (4 * norm z) ≤ of_nat (Suc n) / (2 * norm t)"
by (intro divide_left_mono mult_pos_pos) simp_all
also have "… < norm (of_nat (Suc n) / t) - norm (1 :: 'a)"
using t_nz n by (simp add: field_simps norm_divide del: of_nat_Suc)
also have "… ≤ norm (of_nat (Suc n)/t + 1)" by (rule norm_diff_ineq)
finally have "inverse (norm (of_nat (Suc n)/t + 1)) ≤ 4 * norm z / of_nat (Suc n)"
using z by (simp add: field_split_simps norm_divide mult_ac del: of_nat_Suc)
}
also have "inverse (real_of_nat (Suc n)) * (4 * norm z / real_of_nat (Suc n)) =
4 * norm z * inverse (of_nat (Suc n)^2)"
by (simp add: field_split_simps power2_eq_square del: of_nat_Suc)
finally show "norm (?f n t) ≤ 4 * norm z * inverse (of_nat (Suc n)^2)"
by (simp del: of_nat_Suc)
qed
next
show "summable (λn. 4 * norm z * inverse ((of_nat (Suc n))^2))"
by (subst summable_Suc_iff) (simp add: summable_mult inverse_power_summable)
qed

subsection ‹The Polygamma functions›

lemma summable_deriv_ln_Gamma:
"z ≠ (0 :: 'a :: {real_normed_field,banach}) ⟹
summable (λn. inverse (of_nat (Suc n)) - inverse (z + of_nat (Suc n)))"
unfolding summable_iff_convergent
by (rule uniformly_convergent_imp_convergent,
rule uniformly_summable_deriv_ln_Gamma[of z "norm z/2"]) simp_all

definition✐‹tag important› Polygamma :: "nat ⇒ ('a :: {real_normed_field,banach}) ⇒ 'a" where
"Polygamma n z = (if n = 0 then
(∑k. inverse (of_nat (Suc k)) - inverse (z + of_nat k)) - euler_mascheroni else
(-1)^Suc n * fact n * (∑k. inverse ((z + of_nat k)^Suc n)))"

abbreviation✐‹tag important› Digamma :: "('a :: {real_normed_field,banach}) ⇒ 'a" where
"Digamma ≡ Polygamma 0"

lemma Digamma_def:
"Digamma z = (∑k. inverse (of_nat (Suc k)) - inverse (z + of_nat k)) - euler_mascheroni"

lemma summable_Digamma:
assumes "(z :: 'a :: {real_normed_field,banach}) ≠ 0"
shows   "summable (λn. inverse (of_nat (Suc n)) - inverse (z + of_nat n))"
proof -
have sums: "(λn. inverse (z + of_nat (Suc n)) - inverse (z + of_nat n)) sums
(0 - inverse (z + of_nat 0))"
by (intro telescope_sums filterlim_compose[OF tendsto_inverse_0]
from summable_add[OF summable_deriv_ln_Gamma[OF assms] sums_summable[OF sums]]
show "summable (λn. inverse (of_nat (Suc n)) - inverse (z + of_nat n))" by simp
qed

lemma summable_offset:
assumes "summable (λn. f (n + k) :: 'a :: real_normed_vector)"
shows   "summable f"
proof -
from assms have "convergent (λm. ∑n<m. f (n + k))" by (simp add: summable_iff_convergent)
hence "convergent (λm. (∑n<k. f n) + (∑n<m. f (n + k)))"
also have "(λm. (∑n<k. f n) + (∑n<m. f (n + k))) = (λm. ∑n<m+k. f n)"
proof
fix m :: nat
have "{..<m+k} = {..<k} ∪ {k..<m+k}" by auto
also have "(∑n∈…. f n) = (∑n<k. f n) + (∑n=k..<m+k. f n)"
by (rule sum.union_disjoint) auto
also have "(∑n=k..<m+k. f n) = (∑n=0..<m+k-k. f (n + k))"
using sum.shift_bounds_nat_ivl [of f 0 k m] by simp
finally show "(∑n<k. f n) + (∑n<m. f (n + k)) = (∑n<m+k. f n)" by (simp add: atLeast0LessThan)
qed
finally have "(λa. sum f {..<a}) ⇢ lim (λm. sum f {..<m + k})"
by (auto simp: convergent_LIMSEQ_iff dest: LIMSEQ_offset)
thus ?thesis by (auto simp: summable_iff_convergent convergent_def)
qed

lemma Polygamma_converges:
fixes z :: "'a :: {real_normed_field,banach}"
assumes z: "z ≠ 0" and n: "n ≥ 2"
shows "uniformly_convergent_on (ball z d) (λk z. ∑i<k. inverse ((z + of_nat i)^n))"
proof (rule Weierstrass_m_test'_ev)
define e where "e = (1 + d / norm z)"
define m where "m = nat ⌈norm z * e⌉"
{
fix t assume t: "t ∈ ball z d"
have "norm t = norm (z + (t - z))" by simp
also have "… ≤ norm z + norm (t - z)" by (rule norm_triangle_ineq)
also from t have "norm (t - z) < d" by (simp add: dist_norm norm_minus_commute)
finally have "norm t < norm z * e" using z by (simp add: divide_simps e_def)
} note ball = this

show "eventually (λk. ∀t∈ball z d. norm (inverse ((t + of_nat k)^n)) ≤
inverse (of_nat (k - m)^n)) sequentially"
using eventually_gt_at_top[of m] apply eventually_elim
proof (intro ballI)
fix k :: nat and t :: 'a assume k: "k > m" and t: "t ∈ ball z d"
from k have "real_of_nat (k - m) = of_nat k - of_nat m" by (simp add: of_nat_diff)
also have "… ≤ norm (of_nat k :: 'a) - norm z * e"
unfolding m_def by (subst norm_of_nat) linarith
also from ball[OF t] have "… ≤ norm (of_nat k :: 'a) - norm t" by simp
also have "… ≤ norm (of_nat k + t)" by (rule norm_diff_ineq)
finally have "inverse ((norm (t + of_nat k))^n) ≤ inverse (real_of_nat (k - m)^n)" using k n
thus "norm (inverse ((t + of_nat k)^n)) ≤ inverse (of_nat (k - m)^n)"
by (simp add: norm_inverse norm_power power_inverse)
qed

have "summable (λk. inverse ((real_of_nat k)^n))"
using inverse_power_summable[of n] n by simp
hence "summable (λk. inverse ((real_of_nat (k + m - m))^n))" by simp
thus "summable (λk. inverse ((real_of_nat (k - m))^n))" by (rule summable_offset)
qed

lemma Polygamma_converges':
fixes z :: "'a :: {real_normed_field,banach}"
assumes z: "z ≠ 0" and n: "n ≥ 2"
shows "summable (λk. inverse ((z + of_nat k)^n))"
using uniformly_convergent_imp_convergent[OF Polygamma_converges[OF assms, of 1], of z]

theorem Digamma_LIMSEQ:
fixes z :: "'a :: {banach,real_normed_field}"
assumes z: "z ≠ 0"
shows   "(λm. of_real (ln (real m)) - (∑n<m. inverse (z + of_nat n))) ⇢ Digamma z"
proof -
have "(λn. of_real (ln (real n / (real (Suc n))))) ⇢ (of_real (ln 1) :: 'a)"
by (intro tendsto_intros LIMSEQ_n_over_Suc_n) simp_all
hence "(λn. of_real (ln (real n / (real n + 1)))) ⇢ (0 :: 'a)" by (simp add: add_ac)
hence lim: "(λn. of_real (ln (real n)) - of_real (ln (real n + 1))) ⇢ (0::'a)"
proof (rule Lim_transform_eventually)
show "eventually (λn. of_real (ln (real n / (real n + 1))) =
of_real (ln (real n)) - (of_real (ln (real n + 1)) :: 'a)) at_top"
using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: ln_div)
qed

from summable_Digamma[OF z]
have "(λn. inverse (of_nat (n+1)) - inverse (z + of_nat n))
sums (Digamma z + euler_mascheroni)"
from sums_diff[OF this euler_mascheroni_sum]
have "(λn. of_real (ln (real (Suc n) + 1)) - of_real (ln (real n + 1)) - inverse (z + of_nat n))
hence "(λm. (∑n<m. of_real (ln (real (Suc n) + 1)) - of_real (ln (real n + 1))) -
(∑n<m. inverse (z + of_nat n))) ⇢ Digamma z"
also have "(λm. (∑n<m. of_real (ln (real (Suc n) + 1)) - of_real (ln (real n + 1)))) =
(λm. of_real (ln (m + 1)) :: 'a)"
by (subst sum_lessThan_telescope) simp_all
finally show ?thesis by (rule Lim_transform) (insert lim, simp)
qed

theorem Polygamma_LIMSEQ:
fixes z :: "'a :: {banach,real_normed_field}"
assumes "z ≠ 0" and "n > 0"
shows   "(λk. inverse ((z + of_nat k)^Suc n)) sums ((-1) ^ Suc n * Polygamma n z / fact n)"
using Polygamma_converges'[OF assms(1), of "Suc n"] assms(2)

theorem has_field_derivative_ln_Gamma_complex [derivative_intros]:
fixes z :: complex
assumes z: "z ∉ ℝ⇩≤⇩0"
shows   "(ln_Gamma has_field_derivative Digamma z) (at z)"
proof -
have not_nonpos_Int [simp]: "t ∉ ℤ⇩≤⇩0" if "Re t > 0" for t
using that by (auto elim!: nonpos_Ints_cases')
from z have z': "z ∉ ℤ⇩≤⇩0" and z'': "z ≠ 0" using nonpos_Ints_subset_nonpos_Reals nonpos_Reals_zero_I
by blast+
let ?f' = "λz k. inverse (of_nat (Suc k)) - inverse (z + of_nat (Suc k))"
let ?f = "λz k. z / of_nat (Suc k) - ln (1 + z / of_nat (Suc k))" and ?F' = "λz. ∑n. ?f' z n"
define d where "d = min (norm z/2) (if Im z = 0 then Re z / 2 else abs (Im z) / 2)"
from z have d: "d > 0" "norm z/2 ≥ d" by (auto simp add: complex_nonpos_Reals_iff d_def)
have ball: "Im t = 0 ⟶ Re t > 0" if "dist z t < d" for t
using no_nonpos_Real_in_ball[OF z, of t] that unfolding d_def by (force simp add: complex_nonpos_Reals_iff)
have sums: "(λn. inverse (z + of_nat (Suc n)) - inverse (z + of_nat n)) sums
(0 - inverse (z + of_nat 0))"
by (intro telescope_sums filterlim_compose[OF tendsto_inverse_0]

have "((λz. ∑n. ?f z n) has_field_derivative ?F' z) (at z)"
using d z ln_Gamma_series'_aux[OF z']
apply (intro has_field_derivative_series'(2)[of "ball z d" _ _ z] uniformly_summable_deriv_ln_Gamma)
apply (auto intro!: derivative_eq_intros add_pos_pos mult_pos_pos dest!: ball
simp: field_simps sums_iff nonpos_Reals_divide_of_nat_iff
simp del: of_nat_Suc)
done
with z have "((λz. (∑k. ?f z k) - euler_mascheroni * z - Ln z) has_field_derivative
?F' z - euler_mascheroni - inverse z) (at z)"
by (force intro!: derivative_eq_intros simp: Digamma_def)
also have "?F' z - euler_mascheroni - inverse z = (?F' z + -inverse z) - euler_mascheroni" by simp
also from sums have "-inverse z = (∑n. inverse (z + of_nat (Suc n)) - inverse (z + of_nat n))"
also from sums summable_deriv_ln_Gamma[OF z'']
have "?F' z + … =  (∑n. inverse (of_nat (Suc n)) - inverse (z + of_nat n))"
also have "… - euler_mascheroni = Digamma z" by (simp add: Digamma_def)
finally have "((λz. (∑k. ?f z k) - euler_mascheroni * z - Ln z)
has_field_derivative Digamma z) (at z)" .
moreover from eventually_nhds_ball[OF d(1), of z]
have "eventually (λz. ln_Gamma z = (∑k. ?f z k) - euler_mascheroni * z - Ln z) (nhds z)"
proof eventually_elim
fix t assume "t ∈ ball z d"
hence "t ∉ ℤ⇩≤⇩0" by (auto dest!: ball elim!: nonpos_Ints_cases)
from ln_Gamma_series'_aux[OF this]
show "ln_Gamma t = (∑k. ?f t k) - euler_mascheroni * t - Ln t" by (simp add: sums_iff)
qed
ultimately show ?thesis by (subst DERIV_cong_ev[OF refl _ refl])
qed

declare has_field_derivative_ln_Gamma_complex[THEN DERIV_chain2, derivative_intros]

lemma Digamma_1 [simp]: "Digamma (1 :: 'a :: {real_normed_field,banach}) = - euler_mascheroni"

lemma Digamma_plus1:
assumes "z ≠ 0"
shows   "Digamma (z+1) = Digamma z + 1/z"
proof -
have sums: "(λk. inverse (z + of_nat k) - inverse (z + of_nat (Suc k)))
sums (inverse (z + of_nat 0) - 0)"
by (intro telescope_sums'[OF filterlim_compose[OF tendsto_inverse_0]]
have "Digamma (z+1) = (∑k. inverse (of_nat (Suc k)) - inverse (z + of_nat (Suc k))) -
euler_mascheroni" (is "_ = suminf ?f - _") by (simp add: Digamma_def add_ac)
also have "suminf ?f = (∑k. inverse (of_nat (Suc k)) - inverse (z + of_nat k)) +
(∑k. inverse (z + of_nat k) - inverse (z + of_nat (Suc k)))"
also have "(∑k. inverse (z + of_nat k) - inverse (z + of_nat (Suc k))) = 1/z"
using sums by (simp add: sums_iff inverse_eq_divide)
finally show ?thesis by (simp add: Digamma_def[of z])
qed

theorem Polygamma_plus1:
assumes "z ≠ 0"
shows   "Polygamma n (z + 1) = Polygamma n z + (-1)^n * fact n / (z ^ Suc n)"
proof (cases "n = 0")
assume n: "n ≠ 0"
let ?f = "λk. inverse ((z + of_nat k) ^ Suc n)"
have "Polygamma n (z + 1) = (-1) ^ Suc n * fact n * (∑k. ?f (k+1))"
also have "(∑k. ?f (k+1)) + (∑k<1. ?f k) = (∑k. ?f k)"
using Polygamma_converges'[OF assms, of "Suc n"] n
by (subst suminf_split_initial_segment [symmetric]) simp_all
hence "(∑k. ?f (k+1)) = (∑k. ?f k) - inverse (z ^ Suc n)" by (simp add: algebra_simps)
also have "(-1) ^ Suc n * fact n * ((∑k. ?f k) - inverse (z ^ Suc n)) =
Polygamma n z + (-1)^n * fact n / (z ^ Suc n)" using n
by (simp add: inverse_eq_divide algebra_simps Polygamma_def)
finally show ?thesis .
qed (insert assms, simp add: Digamma_plus1 inverse_eq_divide)

theorem Digamma_of_nat:
"Digamma (of_nat (Suc n) :: 'a :: {real_normed_field,banach}) = harm n - euler_mascheroni"
proof (induction n)
case (Suc n)
have "Digamma (of_nat (Suc (Suc n)) :: 'a) = Digamma (of_nat (Suc n) + 1)" by simp
also have "… = Digamma (of_nat (Suc n)) + inverse (of_nat (Suc n))"
by (subst Digamma_plus1) (simp_all add: inverse_eq_divide del: of_nat_Suc)
also have "Digamma (of_nat (Suc n) :: 'a) = harm n - euler_mascheroni " by (rule Suc)
also have "… + inverse (of_nat (Suc n)) = harm (Suc n) - euler_mascheroni"
finally show ?case .

lemma Digamma_numeral: "Digamma (numeral n) = harm (pred_numeral n) - euler_mascheroni"
by (subst of_nat_numeral[symmetric], subst numeral_eq_Suc, subst Digamma_of_nat) (rule refl)

lemma Polygamma_of_real: "x ≠ 0 ⟹ Polygamma n (of_real x) = of_real (Polygamma n x)"
unfolding Polygamma_def using summable_Digamma[of x] Polygamma_converges'[of x "Suc n"]

lemma Polygamma_Real: "z ∈ ℝ ⟹ z ≠ 0 ⟹ Polygamma n z ∈ ℝ"
by (elim Reals_cases, hypsubst, subst Polygamma_of_real) simp_all

lemma Digamma_half_integer:
"Digamma (of_nat n + 1/2 :: 'a :: {real_normed_field,banach}) =
(∑k<n. 2 / (of_nat (2*k+1))) - euler_mascheroni - of_real (2 * ln 2)"
proof (induction n)
case 0
have "Digamma (1/2 :: 'a) = of_real (Digamma (1/2))" by (simp add: Polygamma_of_real [symmetric])
also have "Digamma (1/2::real) =
(∑k. inverse (of_nat (Suc k)) - inverse (of_nat k + 1/2)) - euler_mascheroni"
also have "(∑k. inverse (of_nat (Suc k) :: real) - inverse (of_nat k + 1/2)) =
(∑k. inverse (1/2) * (inverse (2 * of_nat (Suc k)) - inverse (2 * of_nat k + 1)))"
also have "… = - 2 * ln 2" using sums_minus[OF alternating_harmonic_series_sums']
by (subst suminf_mult) (simp_all add: algebra_simps sums_iff)
finally show ?case by simp
next
case (Suc n)
have nz: "2 * of_nat n + (1:: 'a) ≠ 0"
hence nz': "of_nat n + (1/2::'a) ≠ 0" by (simp add: field_simps)
have "Digamma (of_nat (Suc n) + 1/2 :: 'a) = Digamma (of_nat n + 1/2 + 1)" by simp
also from nz' have "… = Digamma (of_nat n + 1/2) + 1 / (of_nat n + 1/2)"
by (rule Digamma_plus1)
also from nz nz' have "1 / (of_nat n + 1/2 :: 'a) = 2 / (2 * of_nat n + 1)"
by (subst divide_eq_eq) simp_all
also note Suc
qed

lemma Digamma_one_half: "Digamma (1/2) = - euler_mascheroni - of_real (2 * ln 2)"
using Digamma_half_integer[of 0] by simp

lemma Digamma_real_three_halves_pos: "Digamma (3/2 :: real) > 0"
proof -
have "-Digamma (3/2 :: real) = -Digamma (of_nat 1 + 1/2)" by simp
also have "… = 2 * ln 2 + euler_mascheroni - 2" by (subst Digamma_half_integer) simp
also note euler_mascheroni_less_13_over_22
also note ln2_le_25_over_36
finally show ?thesis by simp
qed

theorem has_field_derivative_Polygamma [derivative_intros]:
fixes z :: "'a :: {real_normed_field,euclidean_space}"
assumes z: "z ∉ ℤ⇩≤⇩0"
shows "(Polygamma n has_field_derivative Polygamma (Suc n) z) (at z within A)"
proof (rule has_field_derivative_at_within, cases "n = 0")
assume n: "n = 0"
let ?f = "λk z. inverse (of_nat (Suc k)) - inverse (z + of_nat k)"
let ?F = "λz. ∑k. ?f k z" and ?f' = "λk z. inverse ((z + of_nat k)⇧2)"
from no_nonpos_Int_in_ball'[OF z] guess d . note d = this
from z have summable: "summable (λk. inverse (of_nat (Suc k)) - inverse (z + of_nat k))"
by (intro summable_Digamma) force
from z have conv: "uniformly_convergent_on (ball z d) (λk z. ∑i<k. inverse ((z + of_nat i)⇧2))"
by (intro Polygamma_converges) auto
with d have "summable (λk. inverse ((z + of_nat k)⇧2))" unfolding summable_iff_convergent
by (auto dest!: uniformly_convergent_imp_convergent simp: summable_iff_convergent )

have "(?F has_field_derivative (∑k. ?f' k z)) (at z)"
proof (rule has_field_derivative_series'[of "ball z d" _ _ z])
fix k :: nat and t :: 'a assume t: "t ∈ ball z d"
from t d(2)[of t] show "((λz. ?f k z) has_field_derivative ?f' k t) (at t within ball z d)"
by (auto intro!: derivative_eq_intros simp: power2_eq_square simp del: of_nat_Suc
dest!: plus_of_nat_eq_0_imp elim!: nonpos_Ints_cases)
qed (insert d(1) summable conv, (assumption|simp)+)
with z show "(Polygamma n has_field_derivative Polygamma (Suc n) z) (at z)"
unfolding Digamma_def [abs_def] Polygamma_def [abs_def] using n
by (force simp: power2_eq_square intro!: derivative_eq_intros)
next
assume n: "n ≠ 0"
from z have z': "z ≠ 0" by auto
from no_nonpos_Int_in_ball'[OF z] guess d . note d = this
define n' where "n' = Suc n"
from n have n': "n' ≥ 2" by (simp add: n'_def)
have "((λz. ∑k. inverse ((z + of_nat k) ^ n')) has_field_derivative
(∑k. - of_nat n' * inverse ((z + of_nat k) ^ (n'+1)))) (at z)"
proof (rule has_field_derivative_series'[of "ball z d" _ _ z])
fix k :: nat and t :: 'a assume t: "t ∈ ball z d"
with d have t': "t ∉ ℤ⇩≤⇩0" "t ≠ 0" by auto
show "((λa. inverse ((a + of_nat k) ^ n')) has_field_derivative
- of_nat n' * inverse ((t + of_nat k) ^ (n'+1))) (at t within ball z d)" using t'
by (fastforce intro!: derivative_eq_intros simp: divide_simps power_diff dest: plus_of_nat_eq_0_imp)
next
have "uniformly_convergent_on (ball z d)
(λk z. (- of_nat n' :: 'a) * (∑i<k. inverse ((z + of_nat i) ^ (n'+1))))"
using z' n by (intro uniformly_convergent_mult Polygamma_converges) (simp_all add: n'_def)
thus "uniformly_convergent_on (ball z d)
(λk z. ∑i<k. - of_nat n' * inverse ((z + of_nat i :: 'a) ^ (n'+1)))"
by (subst (asm) sum_distrib_left) simp
qed (insert Polygamma_converges'[OF z' n'] d, simp_all)
also have "(∑k. - of_nat n' * inverse ((z + of_nat k) ^ (n' + 1))) =
(- of_nat n') * (∑k. inverse ((z + of_nat k) ^ (n' + 1)))"
using Polygamma_converges'[OF z', of "n'+1"] n' by (subst suminf_mult) simp_all
finally have "((λz. ∑k. inverse ((z + of_nat k) ^ n')) has_field_derivative
- of_nat n' * (∑k. inverse ((z + of_nat k) ^ (n' + 1)))) (at z)" .
from DERIV_cmult[OF this, of "(-1)^Suc n * fact n :: 'a"]
show "(Polygamma n has_field_derivative Polygamma (Suc n) z) (at z)"
unfolding n'_def Polygamma_def[abs_def] using n by (simp add: algebra_simps)
qed

declare has_field_derivative_Polygamma[THEN DERIV_chain2, derivative_intros]

lemma isCont_Polygamma [continuous_intros]:
fixes f :: "_ ⇒ 'a :: {real_normed_field,euclidean_space}"
shows "isCont f z ⟹ f z ∉ ℤ⇩≤⇩0 ⟹ isCont (λx. Polygamma n (f x)) z"
by (rule isCont_o2[OF _  DERIV_isCont[OF has_field_derivative_Polygamma]])

lemma continuous_on_Polygamma:
"A ∩ ℤ⇩≤⇩0 = {} ⟹ continuous_on A (Polygamma n :: _ ⇒ 'a :: {real_normed_field,euclidean_space})"
by (intro continuous_at_imp_continuous_on isCont_Polygamma[OF continuous_ident] ballI) blast

lemma isCont_ln_Gamma_complex [continuous_intros]:
fixes f :: "'a::t2_space ⇒ complex"
shows "isCont f z ⟹ f z ∉ ℝ⇩≤⇩0 ⟹ isCont (λz. ln_Gamma (f z)) z"
by (rule isCont_o2[OF _  DERIV_isCont[OF has_field_derivative_ln_Gamma_complex]])

lemma continuous_on_ln_Gamma_complex [continuous_intros]:
fixes A :: "complex set"
shows "A ∩ ℝ⇩≤⇩0 = {} ⟹ continuous_on A ln_Gamma"
by (intro continuous_at_imp_continuous_on ballI isCont_ln_Gamma_complex[OF continuous_ident])
fastforce

lemma deriv_Polygamma:
assumes "z ∉ ℤ⇩≤⇩0"
shows   "deriv (Polygamma m) z =
Polygamma (Suc m) (z :: 'a :: {real_normed_field,euclidean_space})"
by (intro DERIV_imp_deriv has_field_derivative_Polygamma assms)
thm has_field_derivative_Polygamma

lemma higher_deriv_Polygamma:
assumes "z ∉ ℤ⇩≤⇩0"
shows   "(deriv ^^ n) (Polygamma m) z =
Polygamma (m + n) (z :: 'a :: {real_normed_field,euclidean_space})"
proof -
have "eventually (λu. (deriv ^^ n) (Polygamma m) u = Polygamma (m + n) u) (nhds z)"
proof (induction n)
case (Suc n)
from Suc.IH have "eventually (λz. eventually (λu. (deriv ^^ n) (Polygamma m) u = Polygamma (m + n) u) (nhds z)) (nhds z)"
hence "eventually (λz. deriv ((deriv ^^ n) (Polygamma m)) z =
deriv (Polygamma (m + n)) z) (nhds z)"
by eventually_elim (intro deriv_cong_ev refl)
moreover have "eventually (λz. z ∈ UNIV - ℤ⇩≤⇩0) (nhds z)" using assms
by (intro eventually_nhds_in_open open_Diff open_UNIV) auto
ultimately show ?case by eventually_elim (simp_all add: deriv_Polygamma)
qed simp_all
thus ?thesis by (rule eventually_nhds_x_imp_x)
qed

lemma deriv_ln_Gamma_complex:
assumes "z ∉ ℝ⇩≤⇩0"
shows   "deriv ln_Gamma z = Digamma (z :: complex)"
by (intro DERIV_imp_deriv has_field_derivative_ln_Gamma_complex assms)

text ‹
We define a type class that captures all the fundamental properties of the inverse of the Gamma function
and defines the Gamma function upon that. This allows us to instantiate the type class both for
the reals and for the complex numbers with a minimal amount of proof duplication.
›

class✐‹tag unimportant› Gamma = real_normed_field + complete_space +
fixes rGamma :: "'a ⇒ 'a"
assumes rGamma_eq_zero_iff_aux: "rGamma z = 0 ⟷ (∃n. z = - of_nat n)"
assumes differentiable_rGamma_aux1:
"(⋀n. z ≠ - of_nat n) ⟹
let d = (THE d. (λn. ∑k<n. inverse (of_nat (Suc k)) - inverse (z + of_nat k))
⇢ d) - scaleR euler_mascheroni 1
in  filterlim (λy. (rGamma y - rGamma z + rGamma z * d * (y - z)) /⇩R
norm (y - z)) (nhds 0) (at z)"
assumes differentiable_rGamma_aux2:
"let z = - of_nat n
in  filterlim (λy. (rGamma y - rGamma z - (-1)^n * (prod of_nat {1..n}) * (y - z)) /⇩R
norm (y - z)) (nhds 0) (at z)"
assumes rGamma_series_aux: "(⋀n. z ≠ - of_nat n) ⟹
let fact' = (λn. prod of_nat {1..n});
exp = (λx. THE e. (λn. ∑k<n. x^k /⇩R fact k) ⇢ e);
pochhammer' = (λa n. (∏n = 0..n. a + of_nat n))
in  filterlim (λn. pochhammer' z n / (fact' n * exp (z * (ln (of_nat n) *⇩R 1))))
(nhds (rGamma z)) sequentially"
begin
subclass banach ..
end

definition "Gamma z = inverse (rGamma z)"

subsection ‹Basic properties›

lemma Gamma_nonpos_Int: "z ∈ ℤ⇩≤⇩0 ⟹ Gamma z = 0"
and rGamma_nonpos_Int: "z ∈ ℤ⇩≤⇩0 ⟹ rGamma z = 0"
using rGamma_eq_zero_iff_aux[of z] unfolding Gamma_def by (auto elim!: nonpos_Ints_cases')

lemma Gamma_nonzero: "z ∉ ℤ⇩≤⇩0 ⟹ Gamma z ≠ 0"
and rGamma_nonzero: "z ∉ ℤ⇩≤⇩0 ⟹ rGamma z ≠ 0"
using rGamma_eq_zero_iff_aux[of z] unfolding Gamma_def by (auto elim!: nonpos_Ints_cases')

lemma Gamma_eq_zero_iff: "Gamma z = 0 ⟷ z ∈ ℤ⇩≤⇩0"
and rGamma_eq_zero_iff: "rGamma z = 0 ⟷ z ∈ ℤ⇩≤⇩0"
using rGamma_eq_zero_iff_aux[of z] unfolding Gamma_def by (auto elim!: nonpos_Ints_cases')

lemma rGamma_inverse_Gamma: "rGamma z = inverse (Gamma z)"
unfolding Gamma_def by simp

lemma rGamma_series_LIMSEQ [tendsto_intros]:
"rGamma_series z ⇢ rGamma z"
proof (cases "z ∈ ℤ⇩≤⇩0")
case False
hence "z ≠ - of_nat n" for n by auto
from rGamma_series_aux[OF this] show ?thesis
by (simp add: rGamma_series_def[abs_def] fact_prod pochhammer_Suc_prod
exp_def of_real_def[symmetric] suminf_def sums_def[abs_def] atLeast0AtMost)
qed (insert rGamma_eq_zero_iff[of z], simp_all add: rGamma_series_nonpos_Ints_LIMSEQ)

theorem Gamma_series_LIMSEQ [tendsto_intros]:
"Gamma_series z ⇢ Gamma z"
proof (cases "z ∈ ℤ⇩≤⇩0")
case False
hence "(λn. inverse (rGamma_series z n)) ⇢ inverse (rGamma z)"
by (intro tendsto_intros) (simp_all add: rGamma_eq_zero_iff)
also have "(λn. inverse (rGamma_series z n)) = Gamma_series z"
finally show ?thesis by (simp add: Gamma_def)
qed (insert Gamma_eq_zero_iff[of z], simp_all add: Gamma_series_nonpos_Ints_LIMSEQ)

lemma Gamma_altdef: "Gamma z = lim (Gamma_series z)"
using Gamma_series_LIMSEQ[of z] by (simp add: limI)

lemma rGamma_1 [simp]: "rGamma 1 = 1"
proof -
have A: "eventually (λn. rGamma_series 1 n = of_nat (Suc n) / of_nat n) sequentially"
using eventually_gt_at_top[of "0::nat"]
by (force elim!: eventually_mono simp: rGamma_series_def exp_of_real pochhammer_fact
field_split_simps pochhammer_rec' dest!: pochhammer_eq_0_imp_nonpos_Int)
have "rGamma_series 1 ⇢ 1" by (subst tendsto_cong[OF A]) (rule LIMSEQ_Suc_n_over_n)
moreover have "rGamma_series 1 ⇢ rGamma 1" by (rule tendsto_intros)
ultimately show ?thesis by (intro LIMSEQ_unique)
qed

lemma rGamma_plus1: "z * rGamma (z + 1) = rGamma z"
proof -
let ?f = "λn. (z + 1) * inverse (of_nat n) + 1"
have "eventually (λn. ?f n * rGamma_series z n = z * rGamma_series (z + 1) n) sequentially"
using eventually_gt_at_top[of "0::nat"]
proof eventually_elim
fix n :: nat assume n: "n > 0"
hence "z * rGamma_series (z + 1) n = inverse (of_nat n) *
pochhammer z (Suc (Suc n)) / (fact n * exp (z * of_real (ln (of_nat n))))"
also from n have "… = ?f n * rGamma_series z n"
finally show "?f n * rGamma_series z n = z * rGamma_series (z + 1) n" ..
qed
moreover have "(λn. ?f n * rGamma_series z n) ⇢ ((z+1) * 0 + 1) * rGamma z"
by (intro tendsto_intros lim_inverse_n)
hence "(λn. ?f n * rGamma_series z n) ⇢ rGamma z" by simp
ultimately have "(λn. z * rGamma_series (z + 1) n) ⇢ rGamma z"
by (blast intro: Lim_transform_eventually)
moreover have "(λn. z * rGamma_series (z + 1) n) ⇢ z * rGamma (z + 1)"
by (intro tendsto_intros)
ultimately show "z * rGamma (z + 1) = rGamma z" using LIMSEQ_unique by blast
qed

lemma pochhammer_rGamma: "rGamma z = pochhammer z n * rGamma (z + of_nat n)"
proof (induction n arbitrary: z)
case (Suc n z)
have "rGamma z = pochhammer z n * rGamma (z + of_nat n)" by (rule Suc.IH)
also note rGamma_plus1 [symmetric]
qed simp_all

theorem Gamma_plus1: "z ∉ ℤ⇩≤⇩0 ⟹ Gamma (z + 1) = z * Gamma z"
using rGamma_plus1[of z] by (simp add: rGamma_inverse_Gamma field_simps Gamma_eq_zero_iff)

theorem pochhammer_Gamma: "z ∉ ℤ⇩≤⇩0 ⟹ pochhammer z n = Gamma (z + of_nat n) / Gamma z"
using pochhammer_rGamma[of z]
by (simp add: rGamma_inverse_Gamma Gamma_eq_zero_iff field_simps)

lemma Gamma_0 [simp]: "Gamma 0 = 0"
and rGamma_0 [simp]: "rGamma 0 = 0"
and Gamma_neg_1 [simp]: "Gamma (- 1) = 0"
and rGamma_neg_1 [simp]: "rGamma (- 1) = 0"
and Gamma_neg_numeral [simp]: "Gamma (- numeral n) = 0"
and rGamma_neg_numeral [simp]: "rGamma (- numeral n) = 0"
and Gamma_neg_of_nat [simp]: "Gamma (- of_nat m) = 0"
and rGamma_neg_of_nat [simp]: "rGamma (- of_nat m) = 0"

lemma Gamma_1 [simp]: "Gamma 1 = 1" unfolding Gamma_def by simp

theorem Gamma_fact: "Gamma (1 + of_nat n) = fact n"
by (simp add: pochhammer_fact pochhammer_Gamma of_nat_in_nonpos_Ints_iff flip: of_nat_Suc)

lemma Gamma_numeral: "Gamma (numeral n) = fact (pred_numeral n)"
by (subst of_nat_numeral[symmetric], subst numeral_eq_Suc,
subst of_nat_Suc, subst Gamma_fact) (rule refl)

lemma Gamma_of_int: "Gamma (of_int n) = (if n > 0 then fact (nat (n - 1)) else 0)"
proof (cases "n > 0")
case True
hence "Gamma (of_int n) = Gamma (of_nat (Suc (nat (n - 1))))" by (subst of_nat_Suc) simp_all
with True show ?thesis by (subst (asm) of_nat_Suc, subst (asm) Gamma_fact) simp

lemma rGamma_of_int: "rGamma (of_int n) = (if n > 0 then inverse (fact (nat (n - 1))) else 0)"

lemma Gamma_seriesI:
assumes "(λn. g n / Gamma_series z n) ⇢ 1"
shows   "g ⇢ Gamma z"
proof (rule Lim_transform_eventually)
have "1/2 > (0::real)" by simp
from tendstoD[OF assms, OF this]
show "eventually (λn. g n / Gamma_series z n * Gamma_series z n = g n) sequentially"
by (force elim!: eventually_mono simp: dist_real_def dist_0_norm)
from assms have "(λn. g n / Gamma_series z n * Gamma_series z n) ⇢ 1 * Gamma z"
by (intro tendsto_intros)
thus "(λn. g n / Gamma_series z n * Gamma_series z n) ⇢ Gamma z" by simp
qed

lemma Gamma_seriesI':
assumes "f ⇢ rGamma z"
assumes "(λn. g n * f n) ⇢ 1"
assumes "z ∉ ℤ⇩≤⇩0"
shows   "g ⇢ Gamma z"
proof (rule Lim_transform_eventually)
have "1/2 > (0::real)" by simp
from tendstoD[OF assms(2), OF this] show "eventually (λn. g n * f n / f n = g n) sequentially"
by (force elim!: eventually_mono simp: dist_real_def dist_0_norm)
from assms have "(λn. g n * f n / f n) ⇢ 1 / rGamma z"
by (intro tendsto_divide assms) (simp_all add: rGamma_eq_zero_iff)
thus "(λn. g n * f n / f n) ⇢ Gamma z" by (simp add: Gamma_def divide_inverse)
qed

lemma Gamma_series'_LIMSEQ: "Gamma_series' z ⇢ Gamma z"
by (cases "z ∈ ℤ⇩≤⇩0") (simp_all add: Gamma_nonpos_Int Gamma_seriesI[OF Gamma_series_Gamma_series']
Gamma_series'_nonpos_Ints_LIMSEQ[of z])

subsection ‹Differentiability›

lemma has_field_derivative_rGamma_no_nonpos_int:
assumes "z ∉ ℤ⇩≤⇩0"
shows   "(rGamma has_field_derivative -rGamma z * Digamma z) (at z within A)"
proof (rule has_field_derivative_at_within)
from assms have "z ≠ - of_nat n" for n by auto
from differentiable_rGamma_aux1[OF this]
show "(rGamma has_field_derivative -rGamma z * Digamma z) (at z)"
unfolding Digamma_def suminf_def sums_def[abs_def]
has_field_derivative_def has_derivative_def netlimit_at
by (simp add: Let_def bounded_linear_mult_right mult_ac of_real_def [symmetric])
qed

lemma has_field_derivative_rGamma_nonpos_int:
"(rGamma has_field_derivative (-1)^n * fact n) (at (- of_nat n) within A)"
apply (rule has_field_derivative_at_within)
using differentiable_rGamma_aux2[of n]
unfolding Let_def has_field_derivative_def has_derivative_def netlimit_at
by (simp only: bounded_linear_mult_right mult_ac of_real_def [symmetric] fact_prod) simp

lemma has_field_derivative_rGamma [derivative_intros]:
"(rGamma has_field_derivative (if z ∈ ℤ⇩≤⇩0 then (-1)^(nat ⌊norm z⌋) * fact (nat ⌊norm z⌋)
else -rGamma z * Digamma z)) (at z within A)"
using has_field_derivative_rGamma_no_nonpos_int[of z A]
has_field_derivative_rGamma_nonpos_int[of "nat ⌊norm z⌋" A]
by (auto elim!: nonpos_Ints_cases')

declare has_field_derivative_rGamma_no_nonpos_int [THEN DERIV_chain2, derivative_intros]
declare has_field_derivative_rGamma [THEN DERIV_chain2, derivative_intros]
declare has_field_derivative_rGamma_nonpos_int [derivative_intros]
declare has_field_derivative_rGamma_no_nonpos_int [derivative_intros]
declare has_field_derivative_rGamma [derivative_intros]

theorem has_field_derivative_Gamma [derivative_intros]:
"z ∉ ℤ⇩≤⇩0 ⟹ (Gamma has_field_derivative Gamma z * Digamma z) (at z within A)"
unfolding Gamma_def [abs_def]
by (fastforce intro!: derivative_eq_intros simp: rGamma_eq_zero_iff)

declare has_field_derivative_Gamma[THEN DERIV_chain2, derivative_intros]

(* TODO: Hide ugly facts properly *)
hide_fact rGamma_eq_zero_iff_aux differentiable_rGamma_aux1 differentiable_rGamma_aux2
differentiable_rGamma_aux2 rGamma_series_aux Gamma_class.rGamma_eq_zero_iff_aux

lemma continuous_on_rGamma [continuous_intros]: "continuous_on A rGamma"
by (rule DERIV_continuous_on has_field_derivative_rGamma)+

lemma continuous_on_Gamma [continuous_intros]: "A ∩ ℤ⇩≤⇩0 = {} ⟹ continuous_on A Gamma"
by (rule DERIV_continuous_on has_field_derivative_Gamma)+ blast

lemma isCont_rGamma [continuous_intros]:
"isCont f z ⟹ isCont (λx. rGamma (f x)) z"
by (rule isCont_o2[OF _  DERIV_isCont[OF has_field_derivative_rGamma]])

lemma isCont_Gamma [continuous_intros]:
"isCont f z ⟹ f z ∉ ℤ⇩≤⇩0 ⟹ isCont (λx. Gamma (f x)) z"
by (rule isCont_o2[OF _  DERIV_isCont[OF has_field_derivative_Gamma]])

subsection✐‹tag unimportant› ‹The complex Gamma function›

instantiation✐‹tag unimportant› complex :: Gamma
begin

definition✐‹tag unimportant› rGamma_complex :: "complex ⇒ complex" where
"rGamma_complex z = lim (rGamma_series z)"

lemma rGamma_series_complex_converges:
"convergent (rGamma_series (z :: complex))" (is "?thesis1")
and rGamma_complex_altdef:
"rGamma z = (if z ∈ ℤ⇩≤⇩0 then 0 else exp (-ln_Gamma z))" (is "?thesis2")
proof -
have "?thesis1 ∧ ?thesis2"
proof (cases "z ∈ ℤ⇩≤⇩0")
case False
have "rGamma_series z ⇢ exp (- ln_Gamma z)"
proof (rule Lim_transform_eventually)
from ln_Gamma_series_complex_converges'[OF False] guess d by (elim exE conjE)
from this(1) uniformly_convergent_imp_convergent[OF this(2), of z]
have "ln_Gamma_series z ⇢ lim (ln_Gamma_series z)" by (simp add: convergent_LIMSEQ_iff)
thus "(λn. exp (-ln_Gamma_series z n)) ⇢ exp (- ln_Gamma z)"
unfolding convergent_def ln_Gamma_def by (intro tendsto_exp tendsto_minus)
from eventually_gt_at_top[of "0::nat"] exp_ln_Gamma_series_complex False
show "eventually (λn. exp (-ln_Gamma_series z n) = rGamma_series z n) sequentially"
by (force elim!: eventually_mono simp: exp_minus Gamma_series_def rGamma_series_def)
qed
with False show ?thesis
by (auto simp: convergent_def rGamma_complex_def intro!: limI)
next
case True
then obtain k where "z = - of_nat k" by (erule nonpos_Ints_cases')
also have "rGamma_series … ⇢ 0"
by (subst tendsto_cong[OF rGamma_series_minus_of_nat]) (simp_all add: convergent_const)
finally show ?thesis using True
by (auto simp: rGamma_complex_def convergent_def intro!: limI)
qed
thus "?thesis1" "?thesis2" by blast+
qed

context✐‹tag unimportant›
begin

(* TODO: duplication *)
private lemma rGamma_complex_plus1: "z * rGamma (z + 1) = rGamma (z :: complex)"
proof -
let ?f = "λn. (z + 1) * inverse (of_nat n) + 1"
have "eventually (λn. ?f n * rGamma_series z n = z * rGamma_series (z + 1) n) sequentially"
using eventually_gt_at_top[of "0::nat"]
proof eventually_elim
fix n :: nat assume n: "n > 0"
hence "z * rGamma_series (z + 1) n = inverse (of_nat n) *
pochhammer z (Suc (Suc n)) / (fact n * exp (z * of_real (ln (of_nat n))))"
also from n have "… = ?f n * rGamma_series z n"
finally show "?f n * rGamma_series z n = z * rGamma_series (z + 1) n" ..
qed
moreover have "(λn. ?f n * rGamma_series z n) ⇢ ((z+1) * 0 + 1) * rGamma z"
using rGamma_series_complex_converges
by (intro tendsto_intros lim_inverse_n)
hence "(λn. ?f n * rGamma_series z n) ⇢ rGamma z" by simp
ultimately have "(λn. z * rGamma_series (z + 1) n) ⇢ rGamma z"
by (blast intro: Lim_transform_eventually)
moreover have "(λn. z * rGamma_series (z + 1) n) ⇢ z * rGamma (z + 1)"
using rGamma_series_complex_converges
by (auto intro!: tendsto_mult simp: rGamma_complex_def convergent_LIMSEQ_iff)
ultimately show "z * rGamma (z + 1) = rGamma z" using LIMSEQ_unique by blast
qed

private lemma has_field_derivative_rGamma_complex_no_nonpos_Int:
assumes "(z :: complex) ∉ ℤ⇩≤⇩0"
shows   "(rGamma has_field_derivative - rGamma z * Digamma z) (at z)"
proof -
have diff: "(rGamma has_field_derivative - rGamma z * Digamma z) (at z)" if "Re z > 0" for z
proof (subst DERIV_cong_ev[OF refl _ refl])
from that have "eventually (λt. t ∈ ball z (Re z/2)) (nhds z)"
by (intro eventually_nhds_in_nhd) simp_all
thus "eventually (λt. rGamma t = exp (- ln_Gamma t)) (nhds z)"
using no_nonpos_Int_in_ball_complex[OF that]
by (auto elim!: eventually_mono simp: rGamma_complex_altdef)
next
have "z ∉ ℝ⇩≤⇩0" using that by (simp add: complex_nonpos_Reals_iff)
with that show "((λt. exp (- ln_Gamma t)) has_field_derivative (-rGamma z * Digamma z)) (at z)"
by (force elim!: nonpos_Ints_cases intro!: derivative_eq_intros simp: rGamma_complex_altdef)
qed

from assms show "(rGamma has_field_derivative - rGamma z * Digamma z) (at z)"
proof (induction "nat ⌊1 - Re z⌋" arbitrary: z)
case (Suc n z)
from Suc.prems have z: "z ≠ 0" by auto
from Suc.hyps have "n = nat ⌊- Re z⌋" by linarith
hence A: "n = nat ⌊1 - Re (z + 1)⌋" by simp
from Suc.prems have B: "z + 1 ∉ ℤ⇩≤⇩0" by (force dest: plus_one_in_nonpos_Ints_imp)

have "((λz. z * (rGamma ∘ (λz. z + 1)) z) has_field_derivative
-rGamma (z + 1) * (Digamma (z + 1) * z - 1)) (at z)"
by (rule derivative_eq_intros DERIV_chain Suc refl A B)+ (simp add: algebra_simps)
also have "(λz. z * (rGamma ∘ (λz. z + 1 :: complex)) z) = rGamma"
also from z have "Digamma (z + 1) * z - 1 = z * Digamma z"
by (subst Digamma_plus1) (simp_all add: field_simps)
also have "-rGamma (z + 1) * (z * Digamma z) = -rGamma z * Digamma z"
by (simp add: rGamma_complex_plus1[of z, symmetric])
finally show ?case .
qed (intro diff, simp)
qed

private lemma rGamma_complex_1: "rGamma (1 :: complex) = 1"
proof -
have A: "eventually (λn. rGamma_series 1 n = of_nat (Suc n) / of_nat n) sequentially"
using eventually_gt_at_top[of "0::nat"]
by (force elim!: eventually_mono simp: rGamma_series_def exp_of_real pochhammer_fact
field_split_simps pochhammer_rec' dest!: pochhammer_eq_0_imp_nonpos_Int)
have "rGamma_series 1 ⇢ 1" by (subst tendsto_cong[OF A]) (rule LIMSEQ_Suc_n_over_n)
thus "rGamma 1 = (1 :: complex)" unfolding rGamma_complex_def by (rule limI)
qed

private lemma has_field_derivative_rGamma_complex_nonpos_Int:
"(rGamma has_field_derivative (-1)^n * fact n) (at (- of_nat n :: complex))"
proof (induction n)
case 0
have A: "(0::complex) + 1 ∉ ℤ⇩≤⇩0" by simp
have "((λz. z * (rGamma ∘ (λz. z + 1 :: complex)) z) has_field_derivative 1) (at 0)"
by (rule derivative_eq_intros DERIV_chain refl
thus ?case by (simp add: rGamma_complex_plus1)
next
case (Suc n)
hence A: "(rGamma has_field_derivative (-1)^n * fact n)
(at (- of_nat (Suc n) + 1 :: complex))" by simp
have "((λz. z * (rGamma ∘ (λz. z + 1 :: complex)) z) has_field_derivative
(- 1) ^ Suc n * fact (Suc n)) (at (- of_nat (Suc n)))"
by (rule derivative_eq_intros refl A DERIV_chain)+
thus ?case by (simp add: rGamma_complex_plus1)
qed

instance proof
fix z :: complex show "(rGamma z = 0) ⟷ (∃n. z = - of_nat n)"
by (auto simp: rGamma_complex_altdef elim!: nonpos_Ints_cases')
next
fix z :: complex assume "⋀n. z ≠ - of_nat n"
hence "z ∉ ℤ⇩≤⇩0" by (auto elim!: nonpos_Ints_cases')
from has_field_derivative_rGamma_complex_no_nonpos_Int[OF this]
show "let d = (THE d. (λn. ∑k<n. inverse (of_nat (Suc k)) - inverse (z + of_nat k))
⇢ d) - euler_mascheroni *⇩R 1 in  (λy. (rGamma y - rGamma z +
rGamma z * d * (y - z)) /⇩R  cmod (y - z)) ─z→ 0"
by (simp add: has_field_derivative_def has_derivative_def Digamma_def sums_def [abs_def]
netlimit_at of_real_def[symmetric] suminf_def)
next
fix n :: nat
from has_field_derivative_rGamma_complex_nonpos_Int[of n]
show "let z = - of_nat n in (λy. (rGamma y - rGamma z - (- 1) ^ n * prod of_nat {1..n} *
(y - z)) /⇩R cmod (y - z)) ─z→ 0"
by (simp add: has_field_derivative_def has_derivative_def fact_prod netlimit_at Let_def)
next
fix z :: complex
from rGamma_series_complex_converges[of z] have "rGamma_series z ⇢ rGamma z"
thus "let fact' = λn. prod of_nat {1..n};
exp = λx. THE e. (λn. ∑k<n. x ^ k /⇩R fact k) ⇢ e;
pochhammer' = λa n. ∏n = 0..n. a + of_nat n
in  (λn. pochhammer' z n / (fact' n * exp (z * ln (real_of_nat n) *⇩R 1))) ⇢ rGamma z"
by (simp add: fact_prod pochhammer_Suc_prod rGamma_series_def [abs_def] exp_def
of_real_def [symmetric] suminf_def sums_def [abs_def] atLeast0AtMost)
qed

end
end

lemma Gamma_complex_altdef:
"Gamma z = (if z ∈ ℤ⇩≤⇩0 then 0 else exp (ln_Gamma (z :: complex)))"
unfolding Gamma_def rGamma_complex_altdef by (simp add: exp_minus)

lemma cnj_rGamma: "cnj (rGamma z) = rGamma (cnj z)"
proof -
have "rGamma_series (cnj z) = (λn. cnj (rGamma_series z n))"
by (intro ext) (simp_all add: rGamma_series_def exp_cnj)
also have "... ⇢ cnj (rGamma z)" by (intro tendsto_cnj tendsto_intros)
finally show ?thesis unfolding rGamma_complex_def by (intro sym[OF limI])
qed

lemma cnj_Gamma: "cnj (Gamma z) = Gamma (cnj z)"
unfolding Gamma_def by (simp add: cnj_rGamma)

lemma Gamma_complex_real:
"z ∈ ℝ ⟹ Gamma z ∈ (ℝ :: complex set)" and rGamma_complex_real: "z ∈ ℝ ⟹ rGamma z ∈ ℝ"
by (simp_all add: Reals_cnj_iff cnj_Gamma cnj_rGamma)

lemma field_differentiable_rGamma: "rGamma field_differentiable (at z within A)"
using has_field_derivative_rGamma[of z] unfolding field_differentiable_def by blast

lemma holomorphic_rGamma [holomorphic_intros]: "rGamma holomorphic_on A"
unfolding holomorphic_on_def by (auto intro!: field_differentiable_rGamma)

lemma holomorphic_rGamma' [holomorphic_intros]:
assumes "f holomorphic_on A"
shows   "(λx. rGamma (f x)) holomorphic_on A"
proof -
have "rGamma ∘ f holomorphic_on A" using assms
by (intro holomorphic_on_compose assms holomorphic_rGamma)
thus ?thesis by (simp only: o_def)
qed

lemma analytic_rGamma: "rGamma analytic_on A"
unfolding analytic_on_def by (auto intro!: exI[of _ 1] holomorphic_rGamma)

lemma field_differentiable_Gamma: "z ∉ ℤ⇩≤⇩0 ⟹ Gamma field_differentiable (at z within A)"
using has_field_derivative_Gamma[of z] unfolding field_differentiable_def by auto

lemma holomorphic_Gamma [holomorphic_intros]: "A ∩ ℤ⇩≤⇩0 = {} ⟹ Gamma holomorphic_on A"
unfolding holomorphic_on_def by (auto intro!: field_differentiable_Gamma)

lemma holomorphic_Gamma' [holomorphic_intros]:
assumes "f holomorphic_on A" and "⋀x. x ∈ A ⟹ f x ∉ ℤ⇩≤⇩0"
shows   "(λx. Gamma (f x)) holomorphic_on A"
proof -
have "Gamma ∘ f holomorphic_on A" using assms
by (intro holomorphic_on_compose assms holomorphic_Gamma) auto
thus ?thesis by (simp only: o_def)
qed

lemma analytic_Gamma: "A ∩ ℤ⇩≤⇩0 = {} ⟹ Gamma analytic_on A"
by (rule analytic_on_subset[of _ "UNIV - ℤ⇩≤⇩0"], subst analytic_on_open)
(auto intro!: holomorphic_Gamma)

lemma field_differentiable_ln_Gamma_complex:
"z ∉ ℝ⇩≤⇩0 ⟹ ln_Gamma field_differentiable (at (z::complex) within A)"
by (rule field_differentiable_within_subset[of _ _ UNIV])
(force simp: field_differentiable_def intro!: derivative_intros)+

lemma holomorphic_ln_Gamma [holomorphic_intros]: "A ∩ ℝ⇩≤⇩0 = {} ⟹ ln_Gamma holomorphic_on A"
unfolding holomorphic_on_def by (auto intro!: field_differentiable_ln_Gamma_complex)

lemma analytic_ln_Gamma: "A ∩ ℝ⇩≤⇩0 = {} ⟹ ln_Gamma analytic_on A"
by (rule analytic_on_subset[of _ "UNIV - ℝ⇩≤⇩0"], subst analytic_on_open)
(auto intro!: holomorphic_ln_Gamma)

lemma has_field_derivative_rGamma_complex' [derivative_intros]:
"(rGamma has_field_derivative (if z ∈ ℤ⇩≤⇩0 then (-1)^(nat ⌊-Re z⌋) * fact (nat ⌊-Re z⌋) else
-rGamma z * Digamma z)) (at z within A)"
using has_field_derivative_rGamma[of z] by (auto elim!: nonpos_Ints_cases')

declare has_field_derivative_rGamma_complex'[THEN DERIV_chain2, derivative_intros]

lemma field_differentiable_Polygamma:
fixes z :: complex
shows
"z ∉ ℤ⇩≤⇩0 ⟹ Polygamma n field_differentiable (at z within A)"
using has_field_derivative_Polygamma[of z n] unfolding field_differentiable_def by auto

lemma holomorphic_on_Polygamma [holomorphic_intros]: "A ∩ ℤ⇩≤⇩0 = {} ⟹ Polygamma n holomorphic_on A"
unfolding holomorphic_on_def by (auto intro!: field_differentiable_Polygamma)

lemma analytic_on_Polygamma: "A ∩ ℤ⇩≤⇩0 = {} ⟹ Polygamma n analytic_on A"
by (rule analytic_on_subset[of _ "UNIV - ℤ⇩≤⇩0"], subst analytic_on_open)
(auto intro!: holomorphic_on_Polygamma)

subsection✐‹tag unimportant› ‹The real Gamma function›

lemma rGamma_series_real:
"eventually (λn. rGamma_series x n = Re (rGamma_series (of_real x) n)) sequentially"
using eventually_gt_at_top[of "0 :: nat"]
proof eventually_elim
fix n :: nat assume n: "n > 0"
have "Re (rGamma_series (of_real x) n) =
Re (of_real (pochhammer x (Suc n)) / (fact n * exp (of_real (x * ln (real_of_nat n)))))"
using n by (simp add: rGamma_series_def powr_def Ln_of_nat pochhammer_of_real)
also from n have "… = Re (of_real ((pochhammer x (Suc n)) /
(fact n * (exp (x * ln (real_of_nat n))))))"
by (subst exp_of_real) simp
also from n have "… = rGamma_series x n"
by (subst Re_complex_of_real) (simp add: rGamma_series_def powr_def)
finally show "rGamma_series x n = Re (rGamma_series (of_real x) n)" ..
qed

instantiation✐‹tag unimportant› real :: Gamma
begin

definition "rGamma_real x = Re (rGamma (of_real x :: complex))"

instance proof
fix x :: real
have "rGamma x = Re (rGamma (of_real x))" by (simp add: rGamma_real_def)
also have "of_real … = rGamma (of_real x :: complex)"
by (intro of_real_Re rGamma_complex_real) simp_all
also have "… = 0 ⟷ x ∈ ℤ⇩≤⇩0" by (simp add: rGamma_eq_zero_iff of_real_in_nonpos_Ints_iff)
also have "… ⟷ (∃n. x = - of_nat n)" by (auto elim!: nonpos_Ints_cases')
finally show "(rGamma x) = 0 ⟷ (∃n. x = - real_of_nat n)" by simp
next
fix x :: real assume "⋀n. x ≠ - of_nat n"
hence x: "complex_of_real x ∉ ℤ⇩≤⇩0"
by (subst of_real_in_nonpos_Ints_iff) (auto elim!: nonpos_Ints_cases')
then have "x ≠ 0" by auto
with x have "(rGamma has_field_derivative - rGamma x * Digamma x) (at x)"
by (fastforce intro!: derivative_eq_intros has_vector_derivative_real_field
simp: Polygamma_of_real rGamma_real_def [abs_def])
thus "let d = (THE d. (λn. ∑k<n. inverse (of_nat (Suc k)) - inverse (x + of_nat k))
⇢ d) - euler_mascheroni *⇩R 1 in  (λy. (rGamma y - rGamma x +
rGamma x * d * (y - x)) /⇩R  norm (y - x)) ─x→ 0"
by (simp add: has_field_derivative_def has_derivative_def Digamma_def sums_def [abs_def]
netlimit_at of_real_def[symmetric] suminf_def)
next
fix n :: nat
have "(rGamma has_field_derivative (-1)^n * fact n) (at (- of_nat n :: real))"
by (fastforce intro!: derivative_eq_intros has_vector_derivative_real_field
simp: Polygamma_of_real rGamma_real_def [abs_def])
thus "let x = - of_nat n in (λy. (rGamma y - rGamma x - (- 1) ^ n * prod of_nat {1..n} *
(y - x)) /⇩R norm (y - x)) ─x::real→ 0"
by (simp add: has_field_derivative_def has_derivative_def fact_prod netlimit_at Let_def)
next
fix x :: real
have "rGamma_series x ⇢ rGamma x"
proof (rule Lim_transform_eventually)
show "(λn. Re (rGamma_series (of_real x) n)) ⇢ rGamma x" unfolding rGamma_real_def
by (intro tendsto_intros)
qed (insert rGamma_series_real, simp add: eq_commute)
thus "let fact' = λn. prod of_nat {1..n};
exp = λx. THE e. (λn. ∑k<n. x ^ k /⇩R fact k) ⇢ e;
pochhammer' = λa n. ∏n = 0..n. a + of_nat n
in  (λn. pochhammer' x n / (fact' n * exp (x * ln (real_of_nat n) *⇩R 1))) ⇢ rGamma x"
by (simp add: fact_prod pochhammer_Suc_prod rGamma_series_def [abs_def] exp_def
of_real_def [symmetric] suminf_def sums_def [abs_def] atLeast0AtMost)
qed

end

lemma rGamma_complex_of_real: "rGamma (complex_of_real x) = complex_of_real (rGamma x)"
unfolding rGamma_real_def using rGamma_complex_real by simp

lemma Gamma_complex_of_real: "Gamma (complex_of_real x) = complex_of_real (Gamma x)"
unfolding Gamma_def by (simp add: rGamma_complex_of_real)

lemma rGamma_real_altdef: "rGamma x = lim (rGamma_series (x :: real))"
by (rule sym, rule limI, rule tendsto_intros)

lemma Gamma_real_altdef1: "Gamma x = lim (Gamma_series (x :: real))"
by (rule sym, rule limI, rule tendsto_intros)

lemma Gamma_real_altdef2: "Gamma x = Re (Gamma (of_real x))"
using rGamma_complex_real[OF Reals_of_real[of x]]
by (elim Reals_cases)
(simp only: Gamma_def rGamma_real_def of_real_inverse[symmetric] Re_complex_of_real)

lemma ln_Gamma_series_complex_of_real:
"x > 0 ⟹ n > 0 ⟹ ln_Gamma_series (complex_of_real x) n = of_real (ln_Gamma_series x n)"
proof -
assume xn: "x > 0" "n > 0"
have "Ln (complex_of_real x / of_nat k + 1) = of_real (ln (x / of_nat k + 1))" if "k ≥ 1" for k
using that xn by (subst Ln_of_real [symmetric]) (auto intro!: add_nonneg_pos simp: field_simps)
with xn show ?thesis by (simp add: ln_Gamma_series_def Ln_of_nat Ln_of_real)
qed

lemma ln_Gamma_real_converges:
assumes "(x::real) > 0"
shows   "convergent (ln_Gamma_series x)"
proof -
have "(λn. ln_Gamma_series (complex_of_real x) n) ⇢ ln_Gamma (of_real x)" using assms
by (intro ln_Gamma_complex_LIMSEQ) (auto simp: of_real_in_nonpos_Ints_iff)
moreover from eventually_gt_at_top[of "0::nat"]
have "eventually (λn. complex_of_real (ln_Gamma_series x n) =
ln_Gamma_series (complex_of_real x) n) sequentially"
by eventually_elim (simp add: ln_Gamma_series_complex_of_real assms)
ultimately have "(λn. complex_of_real (ln_Gamma_series x n)) ⇢ ln_Gamma (of_real x)"
by (subst tendsto_cong) assumption+
from tendsto_Re[OF this] show ?thesis by (auto simp: convergent_def)
qed

lemma ln_Gamma_real_LIMSEQ: "(x::real) > 0 ⟹ ln_Gamma_series x ⇢ ln_Gamma x"
using ln_Gamma_real_converges[of x] unfolding ln_Gamma_def by (simp add: convergent_LIMSEQ_iff)

lemma ln_Gamma_complex_of_real: "x > 0 ⟹ ln_Gamma (complex_of_real x) = of_real (ln_Gamma x)"
proof (unfold ln_Gamma_def, rule limI, rule Lim_transform_eventually)
assume x: "x > 0"
show "eventually (λn. of_real (ln_Gamma_series x n) =
ln_Gamma_series (complex_of_real x) n) sequentially"
using eventually_gt_at_top[of "0::nat"]
by eventually_elim (simp add: ln_Gamma_series_complex_of_real x)
qed (intro tendsto_of_real, insert ln_Gamma_real_LIMSEQ[of x], simp add: ln_Gamma_def)

lemma Gamma_real_pos_exp: "x > (0 :: real) ⟹ Gamma x = exp (ln_Gamma x)"
by (auto simp: Gamma_real_altdef2 Gamma_complex_altdef of_real_in_nonpos_Ints_iff
ln_Gamma_complex_of_real exp_of_real)

lemma ln_Gamma_real_pos: "x > 0 ⟹ ln_Gamma x = ln (Gamma x :: real)"
unfolding Gamma_real_pos_exp by simp

lemma ln_Gamma_complex_conv_fact: "n > 0 ⟹ ln_Gamma (of_nat n :: complex) = ln (fact (n - 1))"
using ln_Gamma_complex_of_real[of "real n"] Gamma_fact[of "n - 1", where 'a = real]
by (simp add: ln_Gamma_real_pos of_nat_diff Ln_of_real [symmetric])

lemma ln_Gamma_real_conv_fact: "n > 0 ⟹ ln_Gamma (real n) = ln (fact (n - 1))"
using Gamma_fact[of "n - 1", where 'a = real]
by (simp add: ln_Gamma_real_pos of_nat_diff Ln_of_real [symmetric])

lemma Gamma_real_pos [simp, intro]: "x > (0::real) ⟹ Gamma x > 0"

lemma Gamma_real_nonneg [simp, intro]: "x > (0::real) ⟹ Gamma x ≥ 0"

lemma has_field_derivative_ln_Gamma_real [derivative_intros]:
assumes x: "x > (0::real)"
shows "(ln_Gamma has_field_derivative Digamma x) (at x)"
proof (subst DERIV_cong_ev[OF refl _ refl])
from assms show "((Re ∘ ln_Gamma ∘ complex_of_real) has_field_derivative Digamma x) (at x)"
by (auto intro!: derivative_eq_intros has_vector_derivative_real_field
simp: Polygamma_of_real o_def)
from eventually_nhds_in_nhd[of x "{0<..}"] assms
show "eventually (λy. ln_Gamma y = (Re ∘ ln_Gamma ∘ of_real) y) (nhds x)"
by (auto elim!: eventually_mono simp: ln_Gamma_complex_of_real interior_open)
qed

lemma field_differentiable_ln_Gamma_real:
"x > 0 ⟹ ln_Gamma field_differentiable (at (x::real) within A)"
by (rule field_differentiable_within_subset[of _ _ UNIV])
(auto simp: field_differentiable_def intro!: derivative_intros)+

declare has_field_derivative_ln_Gamma_real[THEN DERIV_chain2, derivative_intros]

lemma deriv_ln_Gamma_real:
assumes "z > 0"
shows   "deriv ln_Gamma z = Digamma (z :: real)"
by (intro DERIV_imp_deriv has_field_derivative_ln_Gamma_real assms)

lemma has_field_derivative_rGamma_real' [derivative_intros]:
"(rGamma has_field_derivative (if x ∈ ℤ⇩≤⇩0 then (-1)^(nat ⌊-x⌋) * fact (nat ⌊-x⌋) else
-rGamma x * Digamma x)) (at x within A)"
using has_field_derivative_rGamma[of x] by (force elim!: nonpos_Ints_cases')

declare has_field_derivative_rGamma_real'[THEN DERIV_chain2, derivative_intros]

lemma Polygamma_real_odd_pos:
assumes "(x::real) ∉ ℤ⇩≤⇩0" "odd n"
shows   "Polygamma n x > 0"
proof -
from assms have "x ≠ 0" by auto
with assms show ?thesis
unfolding Polygamma_def using Polygamma_converges'[of x "Suc n"]
by (auto simp: zero_less_power_eq simp del: power_Suc
dest: plus_of_nat_eq_0_imp intro!: mult_pos_pos suminf_pos)
qed

lemma Polygamma_real_even_neg:
assumes "(x::real) > 0" "n > 0" "even n"
shows   "Polygamma n x < 0"
using assms unfolding Polygamma_def using Polygamma_converges'[of x "Suc n"]
by (auto intro!: mult_pos_pos suminf_pos)

lemma Polygamma_real_strict_mono:
assumes "x > 0" "x < (y::real)" "even n"
shows   "Polygamma n x < Polygamma n y"
proof -
have "∃ξ. x < ξ ∧ ξ < y ∧ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) ξ"
using assms by (intro MVT2 derivative_intros impI allI) (auto elim!: nonpos_Ints_cases)
then guess ξ by (elim exE conjE) note ξ = this
note ξ(3)
also from ξ(1,2) assms have "(y - x) * Polygamma (Suc n) ξ > 0"
by (intro mult_pos_pos Polygamma_real_odd_pos) (auto elim!: nonpos_Ints_cases)
finally show ?thesis by simp
qed

lemma Polygamma_real_strict_antimono:
assumes "x > 0" "x < (y::real)" "odd n"
shows   "Polygamma n x > Polygamma n y"
proof -
have "∃ξ. x < ξ ∧ ξ < y ∧ Polygamma n y - Polygamma n x = (y - x) * Polygamma (Suc n) ξ"
using assms by (intro MVT2 derivative_intros impI allI) (auto elim!: nonpos_Ints_cases)
then guess ξ by (elim exE conjE) note ξ = this
note ξ(3)
also from ξ(1,2) assms have "(y - x) * Polygamma (Suc n) ξ < 0"
by (intro mult_pos_neg Polygamma_real_even_neg) simp_all
finally show ?thesis by simp
qed

lemma Polygamma_real_mono:
assumes "x > 0" "x ≤ (y::real)" "even n"
shows   "Polygamma n x ≤ Polygamma n y"
using Polygamma_real_strict_mono[OF assms(1) _ assms(3), of y] assms(2)
by (cases "x = y") simp_all

lemma Digamma_real_strict_mono: "(0::real) < x ⟹ x < y ⟹ Digamma x < Digamma y"
by (rule Polygamma_real_strict_mono) simp_all

lemma Digamma_real_mono: "(0::real) < x ⟹ x ≤ y ⟹ Digamma x ≤ Digamma y"
by (rule Polygamma_real_mono) simp_all

lemma Digamma_real_ge_three_halves_pos:
assumes "x ≥ 3/2"
shows   "Digamma (x :: real) > 0"
proof -
have "0 < Digamma (3/2 :: real)" by (fact Digamma_real_three_halves_pos)
also from assms have "… ≤ Digamma x" by (intro Polygamma_real_mono) simp_all
finally show ?thesis .
qed

lemma ln_Gamma_real_strict_mono:
assumes "x ≥ 3/2" "x < y"
shows   "ln_Gamma (x :: real) < ln_Gamma y"
proof -
have "∃ξ. x < ξ ∧ ξ < y ∧ ln_Gamma y - ln_Gamma x = (y - x) * Digamma ξ"
using assms by (intro MVT2 derivative_intros impI allI) (auto elim!: nonpos_Ints_cases)
then guess ξ by (elim exE conjE) note ξ = this
note ξ(3)
also from ξ(1,2) assms have "(y - x) * Digamma ξ > 0"
by (intro mult_pos_pos Digamma_real_ge_three_halves_pos) simp_all
finally show ?thesis by simp
qed

lemma Gamma_real_strict_mono:
assumes "x ≥ 3/2" "x < y"
shows   "Gamma (x :: real) < Gamma y"
proof -
from Gamma_real_pos_exp[of x] assms have "Gamma x = exp (ln_Gamma x)" by simp
also have "… < exp (ln_Gamma y)" by (intro exp_less_mono ln_Gamma_real_strict_mono assms)
also from Gamma_real_pos_exp[of y] assms have "… = Gamma y" by simp
finally show ?thesis .
qed

theorem log_convex_Gamma_real: "convex_on {0<..} (ln ∘ Gamma :: real ⇒ real)"
by (rule convex_on_realI[of _ _ Digamma])
(auto intro!: derivative_eq_intros Polygamma_real_mono Gamma_real_pos
simp: o_def Gamma_eq_zero_iff elim!: nonpos_Ints_cases')

subsection ‹The uniqueness of the real Gamma function›

text ‹
The following is a proof of the Bohr--Mollerup theorem, which states that
any log-convex function ‹G› on the positive reals that fulfils ‹G(1) = 1› and
satisfies the functional equation ‹G(x + 1) = x G(x)› must be equal to the
Gamma function.
In principle, if ‹G› is a holomorphic complex function, one could then extend
this from the positive reals to the entire complex plane (minus the non-positive
integers, where the Gamma function is not defined).
›

context✐‹tag unimportant›
fixes G :: "real ⇒ real"
assumes G_1: "G 1 = 1"
assumes G_plus1: "x > 0 ⟹ G (x + 1) = x * G x"
assumes G_pos: "x > 0 ⟹ G x > 0"
assumes log_convex_G: "convex_on {0<..} (ln ∘ G)"
begin

private lemma G_fact: "G (of_nat n + 1) = fact n"
using G_plus1[of "real n + 1" for n]
by (induction n) (simp_all add: G_1 G_plus1)

private definition S :: "real ⇒ real ⇒ real" where
"S x y = (ln (G y) - ln (G x)) / (y - x)"

private lemma S_eq:
"n ≥ 2 ⟹ S (of_nat n) (of_nat n + x) = (ln (G (real n + x)) - ln (fact (n - 1))) / x"

private lemma G_lower:
assumes x: "x > 0" and n: "n ≥ 1"
shows  "Gamma_series x n ≤ G x"
proof -
have "(ln ∘ G) (real (Suc n)) ≤ ((ln ∘ G) (real (Suc n) + x) -
(ln ∘ G) (real (Suc n) - 1)) / (real (Suc n) + x - (real (Suc n) - 1)) *
(real (Suc n) - (real (Suc n) - 1)) + (ln ∘ G) (real (Suc n) - 1)"
using x n by (intro convex_onD_Icc' convex_on_subset[OF log_convex_G]) auto
hence "S (of_nat n) (of_nat (Suc n)) ≤ S (of_nat (Suc n)) (of_nat (Suc n) + x)"
unfolding S_def using x by (simp add: field_simps)
also have "S (of_nat n) (of_nat (Suc n)) = ln (fact n) - ln (fact (n-1))"
unfolding S_def using n
also have "… = ln (fact n / fact (n-1))" by (subst ln_div) simp_all
also from n have "fact n / fact (n - 1) = n" by (cases n) simp_all
finally have "x * ln (real n) + ln (fact n) ≤ ln (G (real (Suc n) + x))"
using x n by (subst (asm) S_eq) (simp_all add: field_simps)
also have "x * ln (real n) + ln (fact n) = ln (exp (x * ln (real n)) * fact n)"
using x by (simp add: ln_mult)
finally have "exp (x * ln (real n)) * fact n ≤ G (real (Suc n) + x)" using x
by (subst (asm) ln_le_cancel_iff) (simp_all add: G_pos)
also have "G (real (Suc n) + x) = pochhammer x (Suc n) * G x"
using G_plus1[of "real (Suc n) + x" for n] G_plus1[of x] x
finally show "Gamma_series x n ≤ G x"
using x by (simp add: field_simps pochhammer_pos Gamma_series_def)
qed

private lemma G_upper:
assumes x: "x > 0" "x ≤ 1" and n: "n ≥ 2"
shows  "G x ≤ Gamma_series x n * (1 + x / real n)"
proof -
have "(ln ∘ G) (real n + x) ≤ ((ln ∘ G) (real n + 1) -
(ln ∘ G) (real n)) / (real n + 1 - (real n)) *
((real n + x) - real n) + (ln ∘ G) (real n)"
using x n by (intro convex_onD_Icc' convex_on_subset[OF log_convex_G]) auto
hence "S (of_nat n) (of_nat n + x) ≤ S (of_nat n) (of_nat n + 1)"
unfolding S_def using x by (simp add: field_simps)
also from n have "S (of_nat n) (of_nat n + 1) = ln (fact n) - ln (fact (n-1))"
also have "… = ln (fact n / (fact (n-1)))" using n by (subst ln_div) simp_all
also from n have "fact n / fact (n - 1) = n" by (cases n) simp_all
finally have "ln (G (real n + x)) ≤ x * ln (real n) + ln (fact (n - 1))"
using x n by (subst (asm) S_eq) (simp_all add: field_simps)
also have "… = ln (exp (x * ln (real n)) * fact (n - 1))" using x
finally have "G (real n + x) ≤ exp (x * ln (real n)) * fact (n - 1)" using x
by (subst (asm) ln_le_cancel_iff) (simp_all add: G_pos)
also have "G (real n + x) = pochhammer x n * G x"
using G_plus1[of "real n + x" for n] x
finally have "G x ≤ exp (x * ln (real n)) * fact (n- 1) / pochhammer x n"
using x by (simp add: field_simps pochhammer_pos)
also from n have "fact (n - 1) = fact n / n" by (cases n) simp_all
also have "exp (x * ln (real n)) * … / pochhammer x n =
Gamma_series x n * (1 + x / real n)" using n x
by (simp add: Gamma_series_def divide_simps pochhammer_Suc)
finally show ?thesis .
qed

private lemma G_eq_Gamma_aux:
assumes x: "x > 0" "x ≤ 1"
shows   "G x = Gamma x"
proof (rule antisym)
show "G x ≥ Gamma x"
proof (rule tendsto_upperbound)
from G_lower[of x] show "eventually (λn. Gamma_series x n ≤ G x) sequentially"
using  x by (auto intro: eventually_mono[OF eventually_ge_at_top[of "1::nat"]])
next
show "G x ≤ Gamma x"
proof (rule tendsto_lowerbound)
have "(λn. Gamma_series x n * (1 + x / real n)) ⇢ Gamma x * (1 + 0)"
by (rule tendsto_intros real_tendsto_divide_at_top
Gamma_series_LIMSEQ filterlim_real_sequentially)+
thus "(λn. Gamma_series x n * (1 + x / real n)) ⇢ Gamma x" by simp
next
from G_upper[of x] show "eventually (λn. Gamma_series x n * (1 + x / real n) ≥ G x) sequentially"
using x by (auto intro: eventually_mono[OF eventually_ge_at_top[of "2::nat"]])
qed simp_all
qed

theorem Gamma_pos_real_unique:
assumes x: "x > 0"
shows   "G x = Gamma x"
proof -
have G_eq: "G (real n + x) = Gamma (real n + x)" if "x ∈ {0<..1}" for n x using that
proof (induction n)
case (Suc n)
from Suc have "x + real n > 0" by simp
hence "x + real n ∉ ℤ⇩≤⇩0" by auto
with Suc show ?case using G_plus1[of "real n + x"] Gamma_plus1[of "real n + x"]

show ?thesis
proof (cases "frac x = 0")
case True
hence "x = of_int (floor x)" by (simp add: frac_def)
with x have x_eq: "x = of_nat (nat (floor x) - 1) + 1" by simp
show ?thesis by (subst (1 2) x_eq, rule G_eq) simp_all
next
case False
from assms have x_eq: "x = of_nat (nat (floor x)) + frac x"
have frac_le_1: "frac x ≤ 1" unfolding frac_def by linarith
show ?thesis
by (subst (1 2) x_eq, rule G_eq, insert False frac_le_1) simp_all
qed
qed

end

subsection ‹The Beta function›

definition Beta where "Beta a b = Gamma a * Gamma b / Gamma (a + b)"

lemma Beta_altdef: "Beta a b = Gamma a * Gamma b * rGamma (a + b)"
by (simp add: inverse_eq_divide Beta_def Gamma_def)

lemma Beta_commute: "Beta a b = Beta b a"
unfolding Beta_def by (simp add: ac_simps)

lemma has_field_derivative_Beta1 [derivative_intros]:
assumes "x ∉ ℤ⇩≤⇩0" "x + y ∉ ℤ⇩≤⇩0"
shows   "((λx. Beta x y) has_field_derivative (Beta x y * (Digamma x - Digamma (x + y))))
(at x within A)" unfolding Beta_altdef
by (rule DERIV_cong, (rule derivative_intros assms)+) (simp add: algebra_simps)

lemma Beta_pole1: "x ∈ ℤ⇩≤⇩0 ⟹ Beta x y = 0"
by (auto simp add: Beta_def elim!: nonpos_Ints_cases')

lemma Beta_pole2: "y ∈ ℤ⇩≤⇩0 ⟹ Beta x y = 0"
by (auto simp add: Beta_def elim!: nonpos_Ints_cases')

lemma Beta_zero: "x + y ∈ ℤ⇩≤⇩0 ⟹ Beta x y = 0"
by (auto simp add: Beta_def elim!: nonpos_Ints_cases')

lemma has_field_derivative_Beta2 [derivative_intros]:
assumes "y ∉ ℤ⇩≤⇩0" "x + y ∉ ℤ⇩≤⇩0"
shows   "((λy. Beta x y) has_field_derivative (Beta x y * (Digamma y - Digamma (x + y))))
(at y within A)"

theorem Beta_plus1_plus1:
assumes "x ∉ ℤ⇩≤⇩0" "y ∉ ℤ⇩≤⇩0"
shows   "Beta (x + 1) y + Beta x (y + 1) = Beta x y"
proof -
have "Beta (x + 1) y + Beta x (y + 1) =
(Gamma (x + 1) * Gamma y + Gamma x * Gamma (y + 1)) * rGamma ((x + y) + 1)"
also have "… = (Gamma x * Gamma y) * ((x + y) * rGamma ((x + y) + 1))"
by (subst assms[THEN Gamma_plus1])+ (simp add: algebra_simps)
also from assms have "… = Beta x y" unfolding Beta_altdef by (subst rGamma_plus1) simp
finally show ?thesis .
qed

theorem Beta_plus1_left:
assumes "x ∉ ℤ⇩≤⇩0"
shows   "(x + y) * Beta (x + 1) y = x * Beta x y"
proof -
have "(x + y) * Beta (x + 1) y = Gamma (x + 1) * Gamma y * ((x + y) * rGamma ((x + y) + 1))"
unfolding Beta_altdef by (simp only: ac_simps)
also have "… = x * Beta x y" unfolding Beta_altdef
by (subst assms[THEN Gamma_plus1] rGamma_plus1)+ (simp only: ac_simps)
finally show ?thesis .
qed

theorem Beta_plus1_right:
assumes "y ∉ ℤ⇩≤⇩0"
shows   "(x + y) * Beta x (y + 1) = y * Beta x y"

lemma Gamma_Gamma_Beta:
assumes "x + y ∉ ℤ⇩≤⇩0"
shows   "Gamma x * Gamma y = Beta x y * Gamma (x + y)"
unfolding Beta_altdef using assms Gamma_eq_zero_iff[of "x+y"]

subsection ‹Legendre duplication theorem›

context
begin

private lemma Gamma_legendre_duplication_aux:
fixes z :: "'a :: Gamma"
assumes "z ∉ ℤ⇩≤⇩0" "z + 1/2 ∉ ℤ⇩≤⇩0"
shows "Gamma z * Gamma (z + 1/2) = exp ((1 - 2*z) * of_real (ln 2)) * Gamma (1/2) * Gamma (2*z)"
proof -
let ?powr = "λb a. exp (a * of_real (ln (of_nat b)))"
let ?h = "λn. (fact (n-1))⇧2 / fact (2*n-1) * of_nat (2^(2*n)) *
exp (1/2 * of_real (ln (real_of_nat n)))"
{
fix z :: 'a assume z: "z ∉ ℤ⇩≤⇩0" "z + 1/2 ∉ ℤ⇩≤⇩0"
let ?g = "λn. ?powr 2 (2*z) * Gamma_series' z n * Gamma_series' (z + 1/2) n /
Gamma_series' (2*z) (2*n)"
have "eventually (λn. ?g n = ?h n) sequentially" using eventually_gt_at_top
proof eventually_elim
fix n :: nat assume n: "n > 0"
let ?f = "fact (n - 1) :: 'a" and ?f' = "fact (2*n - 1) :: 'a"
have A: "exp t * exp t = exp (2*t :: 'a)" for t by (subst exp_add [symmetric]) simp
have A: "Gamma_series' z n * Gamma_series' (z + 1/2) n = ?f^2 * ?powr n (2*z + 1/2) /
(pochhammer z n * pochhammer (z + 1/2) n)"
have B: "Gamma_series' (2*z) (2*n) =
?f' * ?powr 2 (2*z) * ?powr n (2*z) /
(of_nat (2^(2*n)) * pochhammer z n * pochhammer (z+1/2) n)" using n
from z have "pochhammer z n ≠ 0" by (auto dest: pochhammer_eq_0_imp_nonpos_Int)
moreover from z have "pochhammer (z + 1/2) n ≠ 0" by (auto dest: pochhammer_eq_0_imp_nonpos_Int)
ultimately have "?powr 2 (2*z) * (Gamma_series' z n * Gamma_series' (z + 1/2) n) / Gamma_series' (2*z) (2*n) =
?f^2 / ?f' * of_nat (2^(2*n)) * (?powr n ((4*z + 1)/2) * ?powr n (-2*z))"
using n unfolding A B by (simp add: field_split_simps exp_minus)
also have "?powr n ((4*z + 1)/2) * ?powr n (-2*z) = ?powr n (1/2)"
finally show "?g n = ?h n" by (simp only: mult_ac)
qed

moreover from z double_in_nonpos_Ints_imp[of z] have "2 * z ∉ ℤ⇩≤⇩0" by auto
hence "?g ⇢ ?powr 2 (2*z) * Gamma z * Gamma (z+1/2) / Gamma (2*z)"
using LIMSEQ_subseq_LIMSEQ[OF Gamma_series'_LIMSEQ, of "(*)2" "2*z"]
by (intro tendsto_intros Gamma_series'_LIMSEQ)
ultimately have "?h ⇢ ?powr 2 (2*z) * Gamma z * Gamma (z+1/2) / Gamma (2*z)"
by (blast intro: Lim_transform_eventually)
} note lim = this

from assms double_in_nonpos_Ints_imp[of z] have z': "2 * z ∉ ℤ⇩≤⇩0" by auto
from fraction_not_in_ints[of 2 1] have "(1/2 :: 'a) ∉ ℤ⇩≤⇩0"
by (intro not_in_Ints_imp_not_in_nonpos_Ints) simp_all
with lim[of "1/2 :: 'a"] have "?h ⇢ 2 * Gamma (1/2 :: 'a)" by (simp add: exp_of_real)
from LIMSEQ_unique[OF this lim[OF assms]] z' show ?thesis
by (simp add: field_split_simps Gamma_eq_zero_iff ring_distribs exp_diff exp_of_real ac_simps)
qed

text ‹
The following lemma is somewhat annoying. With a little bit of complex analysis
(Cauchy's integral theorem, to be exact), this would be completely trivial. However,
we want to avoid depending on the complex analysis session at this point, so we prove it
the hard way.
›
private lemma Gamma_reflection_aux:
defines "h ≡ λz::complex. if z ∈ ℤ then 0 else
(of_real pi * cot (of_real pi*z) + Digamma z - Digamma (1 - z))"
defines "a ≡ complex_of_real pi"
obtains h' where "continuous_on UNIV h'" "⋀z. (h has_field_derivative (h' z)) (at z)"
proof -
define f where "f n = a * of_real (cos_coeff (n+1) - sin_coeff (n+2))" for n
define F where "F z = (if z = 0 then 0 else (cos (a*z) - sin (a*z)/(a*z)) / z)" for z
define g where "g n = complex_of_real (sin_coeff (n+1))" for n
define G where "G z = (if z = 0 then 1 else sin (a*z)/(a*z))" for z
have a_nz: "a ≠ 0" unfolding a_def by simp

have "(λn. f n * (a*z)^n) sums (F z) ∧ (λn. g n * (a*z)^n) sums (G z)"
if "abs (Re z) < 1" for z
proof (cases "z = 0"; rule conjI)
assume "z ≠ 0"
note z = this that

from z have sin_nz: "sin (a*z) ≠ 0" unfolding a_def by (auto simp: sin_eq_0)
have "(λn. of_real (sin_coeff n) * (a*z)^n) sums (sin (a*z))" using sin_converges[of "a*z"]
from sums_split_initial_segment[OF this, of 1]
have "(λn. (a*z) * of_real (sin_coeff (n+1)) * (a*z)^n) sums (sin (a*z))" by (simp add: mult_ac)
from sums_mult[OF this, of "inverse (a*z)"] z a_nz
have A: "(λn. g n * (a*z)^n) sums (sin (a*z)/(a*z))"
with z show "(λn. g n * (a*z)^n) sums (G z)" by (simp add: G_def)
from A z a_nz sin_nz have g_nz: "(∑n. g n * (a*z)^n) ≠ 0" by (simp add: sums_iff g_def)

have [simp]: "sin_coeff (Suc 0) = 1" by (simp add: sin_coeff_def)
from sums_split_initial_segment[OF sums_diff[OF cos_converges[of "a*z"] A], of 1]
have "(λn. z * f n * (a*z)^n) sums (cos (a*z) - sin (a*z) / (a*z))"
by (simp add: mult_ac scaleR_conv_of_real ring_distribs f_def g_def)
from sums_mult[OF this, of "inverse z"] z assms
show "(λn. f n * (a*z)^n) sums (F z)" by (simp add: divide_simps mult_ac f_def F_def)
next
assume z: "z = 0"
have "(λn. f n * (a * z) ^ n) sums f 0" using powser_sums_zero[of f] z by simp
with z show "(λn. f n * (a * z) ^ n) sums (F z)"
by (simp add: f_def F_def sin_coeff_def cos_coeff_def)
have "(λn. g n * (a * z) ^ n) sums g 0" using powser_sums_zero[of g] z by simp
with z show "(λn. g n * (a * z) ^ n) sums (G z)"
by (simp add: g_def G_def sin_coeff_def cos_coeff_def)
qed
note sums = conjunct1[OF this] conjunct2[OF this]

define h2 where [abs_def]:
"h2 z = (∑n. f n * (a*z)^n) / (∑n. g n * (a*z)^n) + Digamma (1 + z) - Digamma (1 - z)" for z
define POWSER where [abs_def]: "POWSER f z = (∑n. f n * (z^n :: complex))" for f z
define POWSER' where [abs_def]: "POWSER' f z = (∑n. diffs f n * (z^n))" for f and z :: complex
define h2' where [abs_def]:
"h2' z = a * (POWSER g (a*z) * POWSER' f (a*z) - POWSER f (a*z) * POWSER' g (a*z)) /
(POWSER g (a*z))^2 + Polygamma 1 (1 + z) + Polygamma 1 (1 - z)" for z

have h_eq: "h t = h2 t" if "abs (Re t) < 1" for t
proof -
from that have t: "t ∈ ℤ ⟷ t = 0" by (auto elim!: Ints_cases simp: dist_0_norm)
hence "h t = a*cot (a*t) - 1/t + Digamma (1 + t) - Digamma (1 - t)"
unfolding h_def using Digamma_plus1[of t] by (force simp: field_simps a_def)
also have "a*cot (a*t) - 1/t = (F t) / (G t)"
using t by (auto simp add: divide_simps sin_eq_0 cot_def a_def F_def G_def)
also have "… = (∑n. f n * (a*t)^n) / (∑n. g n * (a*t)^n)"
using sums[of t] that by (simp add: sums_iff dist_0_norm)
finally show "h t = h2 t" by (simp only: h2_def)
qed

let ?A = "{z. abs (Re z) < 1}"
have "open ({z. Re z < 1} ∩ {z. Re z > -1})"
using open_halfspace_Re_gt open_halfspace_Re_lt by auto
also have "({z. Re z < 1} ∩ {z. Re z > -1}) = {z. abs (Re z) < 1}" by auto
finally have open_A: "open ?A" .
hence [simp]: "interior ?A = ?A" by (simp add: interior_open)

have summable_f: "summable (λn. f n * z^n)" for z
by (rule powser_inside, rule sums_summable, rule sums[of "𝗂 * of_real (norm z + 1) / a"])
have summable_g: "summable (λn. g n * z^n)" for z
by (rule powser_inside, rule sums_summable, rule sums[of "𝗂 * of_real (norm z + 1) / a"])
have summable_fg': "summable (λn. diffs f n * z^n)" "summable (λn. diffs g n * z^n)" for z
by (intro termdiff_converges_all summable_f summable_g)+
have "(POWSER f has_field_derivative (POWSER' f z)) (at z)"
"(POWSER g has_field_derivative (POWSER' g z)) (at z)" for z
unfolding POWSER_def POWSER'_def
by (intro termdiffs_strong_converges_everywhere summable_f summable_g)+
note derivs = this[THEN DERIV_chain2[OF _ DERIV_cmult[OF DERIV_ident]], unfolded POWSER_def]
have "isCont (POWSER f) z" "isCont (POWSER g) z" "isCont (POWSER' f) z" "isCont (POWSER' g) z"
for z unfolding POWSER_def POWSER'_def
by (intro isCont_powser_converges_everywhere summable_f summable_g summable_fg')+
note cont = this[THEN isCont_o2[rotated], unfolded POWSER_def POWSER'_def]

{
fix z :: complex assume z: "abs (Re z) < 1"
define d where "d = 𝗂 * of_real (norm z + 1)"
have d: "abs (Re d) < 1" "norm z < norm d" by (simp_all add: d_def norm_mult del: of_real_add)
have "eventually (λz. h z = h2 z) (nhds z)"
using eventually_nhds_in_nhd[of z ?A] using h_eq z
by (auto elim!: eventually_mono simp: dist_0_norm)

moreover from sums(2)[OF z] z have nz: "(∑n. g n * (a * z) ^ n) ≠ 0"
unfolding G_def by (auto simp: sums_iff sin_eq_0 a_def)
have A: "z ∈ ℤ ⟷ z = 0" using z by (auto elim!: Ints_cases)
have no_int: "1 + z ∈ ℤ ⟷ z = 0" using z Ints_diff[of "1+z" 1] A
by (auto elim!: nonpos_Ints_cases)
have no_int': "1 - z ∈ ℤ ⟷ z = 0" using z Ints_diff[of 1 "1-z"] A
by (auto elim!: nonpos_Ints_cases)
from no_int no_int' have no_int: "1 - z ∉ ℤ⇩≤⇩0" "1 + z ∉ ℤ⇩≤⇩0" by auto
have "(h2 has_field_derivative h2' z) (at z)" unfolding h2_def
by (rule DERIV_cong, (rule derivative_intros refl derivs[unfolded POWSER_def] nz no_int)+)
(auto simp: h2'_def POWSER_def field_simps power2_eq_square)
ultimately have deriv: "(h has_field_derivative h2' z) (at z)"
by (subst DERIV_cong_ev[OF refl _ refl])

from sums(2)[OF z] z have "(∑n. g n * (a * z) ^ n) ≠ 0"
unfolding G_def by (auto simp: sums_iff a_def sin_eq_0)
hence "isCont h2' z" using no_int unfolding h2'_def[abs_def] POWSER_def POWSER'_def
by (intro continuous_intros cont
continuous_on_compose2[OF _ continuous_on_Polygamma[of "{z. Re z > 0}"]]) auto
note deriv and this
} note A = this

interpret h: periodic_fun_simple' h
proof
fix z :: complex
show "h (z + 1) = h z"
proof (cases "z ∈ ℤ")
assume z: "z ∉ ℤ"
hence A: "z + 1 ∉ ℤ" "z ≠ 0" using Ints_diff[of "z+1" 1] by auto
hence "Digamma (z + 1) - Digamma (-z) = Digamma z - Digamma (-z + 1)"
by (subst (1 2) Digamma_plus1) simp_all
with A z show "h (z + 1) = h z"
by (simp add: h_def sin_plus_pi cos_plus_pi ring_distribs cot_def)
qed

have h2'_eq: "h2' (z - 1) = h2' z" if z: "Re z > 0" "Re z < 1" for z
proof -
have "((λz. h (z - 1)) has_field_derivative h2' (z - 1)) (at z)"
by (rule DERIV_cong, rule DERIV_chain'[OF _ A(1)])
(insert z, auto intro!: derivative_eq_intros)
hence "(h has_field_derivative h2' (z - 1)) (at z)" by (subst (asm) h.minus_1)
moreover from z have "(h has_field_derivative h2' z) (at z)" by (intro A) simp_all
ultimately show "h2' (z - 1) = h2' z" by (rule DERIV_unique)
qed

define h2'' where "h2'' z = h2' (z - of_int ⌊Re z⌋)" for z
have deriv: "(h has_field_derivative h2'' z) (at z)" for z
proof -
fix z :: complex
have B: "¦Re z - real_of_int ⌊Re z⌋¦ < 1" by linarith
have "((λt. h (t - of_int ⌊Re z⌋)) has_field_derivative h2'' z) (at z)"
unfolding h2''_def by (rule DERIV_cong, rule DERIV_chain'[OF _ A(1)])
(insert B, auto intro!: derivative_intros)
thus "(h has_field_derivative h2'' z) (at z)" by (simp add: h.minus_of_int)
qed

have cont: "continuous_on UNIV h2''"
proof (intro continuous_at_imp_continuous_on ballI)
fix z :: complex
define r where "r = ⌊Re z⌋"
define A where "A = {t. of_int r - 1 < Re t ∧ Re t < of_int r + 1}"
have "continuous_on A (λt. h2' (t - of_int r))" unfolding A_def
by (intro continuous_at_imp_continuous_on isCont_o2[OF _ A(2)] ballI continuous_intros)
moreover have "h2'' t = h2' (t - of_int r)" if t: "t ∈ A" for t
proof (cases "Re t ≥ of_int r")
case True
from t have "of_int r - 1 < Re t" "Re t < of_int r + 1" by (simp_all add: A_def)
with True have "⌊Re t⌋ = ⌊Re z⌋" unfolding r_def by linarith
thus ?thesis by (auto simp: r_def h2''_def)
next
case False
from t have t: "of_int r - 1 < Re t" "Re t < of_int r + 1" by (simp_all add: A_def)
with False have t': "⌊Re t⌋ = ⌊Re z⌋ - 1" unfolding r_def by linarith
moreover from t False have "h2' (t - of_int r + 1 - 1) = h2' (t - of_int r + 1)"
by (intro h2'_eq) simp_all
ultimately show ?thesis by (auto simp: r_def h2''_def algebra_simps t')
qed
ultimately have "continuous_on A h2''" by (subst continuous_on_cong[OF refl])
moreover {
have "open ({t. of_int r - 1 < Re t} ∩ {t. of_int r + 1 > Re t})"
by (intro open_Int open_halfspace_Re_gt open_halfspace_Re_lt)
also have "{t. of_int r - 1 < Re t} ∩ {t. of_int r + 1 > Re t} = A"
unfolding A_def by blast
finally have "open A" .
}
ultimately have C: "isCont h2'' t" if "t ∈ A" for t using that
by (subst (asm) continuous_on_eq_continuous_at) auto
have "of_int r - 1 < Re z" "Re z  < of_int r + 1" unfolding r_def by linarith+
thus "isCont h2'' z" by (intro C) (simp_all add: A_def)
qed

from that[OF cont deriv] show ?thesis .
qed

lemma Gamma_reflection_complex:
fixes z :: complex
shows "Gamma z * Gamma (1 - z) = of_real pi / sin (of_real pi * z)"
proof -
let ?g = "λz::complex. Gamma z * Gamma (1 - z) * sin (of_real pi * z)"
define g where [abs_def]: "g z = (if z ∈ ℤ then of_real pi else ?g z)" for z :: complex
let ?h = "λz::complex. (of_real pi * cot (of_real pi*z) + Digamma z - Digamma (1 - z))"
define h where [abs_def]: "h z = (if z ∈ ℤ then 0 else ?h z)" for z :: complex

― ‹@{term g} is periodic with period 1.›
interpret g: periodic_fun_simple' g
proof
fix z :: complex
show "g (z + 1) = g z"
proof (cases "z ∈ ℤ")
case False
hence "z * g z = z * Beta z (- z + 1) * sin (of_real pi * z)" by (simp add: g_def Beta_def)
also have "z * Beta z (- z + 1) = (z + 1 + -z) * Beta (z + 1) (- z + 1)"
using False Ints_diff[of 1 "1 - z"] nonpos_Ints_subset_Ints
by (subst Beta_plus1_left [symmetric]) auto
also have "… * sin (of_real pi * z) = z * (Beta (z + 1) (-z) * sin (of_real pi * (z + 1)))"
using False Ints_diff[of "z+1" 1] Ints_minus[of "-z"] nonpos_Ints_subset_Ints
by (subst Beta_plus1_right) (auto simp: ring_distribs sin_plus_pi)
also from False have "Beta (z + 1) (-z) * sin (of_real pi * (z + 1)) = g (z + 1)"
using Ints_diff[of "z+1" 1] by (auto simp: g_def Beta_def)
finally show "g (z + 1) = g z" using False by (subst (asm) mult_left_cancel) auto
qed

― ‹@{term g} is entire.›
have g_g': "(g has_field_derivative (h z * g z)) (at z)" for z :: complex
proof (cases "z ∈ ℤ")
let ?h' = "λz. Beta z (1 - z) * ((Digamma z - Digamma (1 - z)) * sin (z * of_real pi) +
of_real pi * cos (z * of_real pi))"
case False
from False have "eventually (λt. t ∈ UNIV - ℤ) (nhds z)"
by (intro eventually_nhds_in_open) (auto simp: open_Diff)
hence "eventually (λt. g t = ?g t) (nhds z)" by eventually_elim (simp add: g_def)
moreover {
from False Ints_diff[of 1 "1-z"] have "1 - z ∉ ℤ" by auto
hence "(?g has_field_derivative ?h' z) (at z)" using nonpos_Ints_subset_Ints
by (auto intro!: derivative_eq_intros simp: algebra_simps Beta_def)
also from False have "sin (of_real pi * z) ≠ 0" by (subst sin_eq_0) auto
hence "?h' z = h z * g z"
using False unfolding g_def h_def cot_def by (simp add: field_simps Beta_def)
finally have "(?g has_field_derivative (h z * g z)) (at z)" .
}
ultimately show ?thesis by (subst DERIV_cong_ev[OF refl _ refl])
next
case True
then obtain n where z: "z = of_int n" by (auto elim!: Ints_cases)
let ?t = "(λz::complex. if z = 0 then 1 else sin z / z) ∘ (λz. of_real pi * z)"
have deriv_0: "(g has_field_derivative 0) (at 0)"
proof (subst DERIV_cong_ev[OF refl _ refl])
show "eventually (λz. g z = of_real pi * Gamma (1 + z) * Gamma (1 - z) * ?t z) (nhds 0)"
using eventually_nhds_ball[OF zero_less_one, of "0::complex"]
proof eventually_elim
fix z :: complex assume z: "z ∈ ball 0 1"
show "g z = of_real pi * Gamma (1 + z) * Gamma (1 - z) * ?t z"
proof (cases "z = 0")
assume z': "z ≠ 0"
with z have z'': "z ∉ ℤ⇩≤⇩0" "z ∉ ℤ" by (auto elim!: Ints_cases simp: dist_0_norm)
from Gamma_plus1[OF this(1)] have "Gamma z = Gamma (z + 1) / z" by simp
with z'' z' show ?thesis by (simp add: g_def ac_simps)
qed
have "(?t has_field_derivative (0 * of_real pi)) (at 0)"
using has_field_derivative_sin_z_over_z[of "UNIV :: complex set"]
by (intro DERIV_chain) simp_all
thus "((λz. of_real pi * Gamma (1 + z) * Gamma (1 - z) * ?t z) has_field_derivative 0) (at 0)"
by (auto intro!: derivative_eq_intros simp: o_def)
qed

have "((g ∘ (λx. x - of_int n)) has_field_derivative 0 * 1) (at (of_int n))"
using deriv_0 by (intro DERIV_chain) (auto intro!: derivative_eq_intros)
also have "g ∘ (λx. x - of_int n) = g" by (intro ext) (simp add: g.minus_of_int)
finally show "(g has_field_derivative (h z * g z)) (at z)" by (simp add: z h_def)
qed

have g_eq: "g (z/2) * g ((z+1)/2) = Gamma (1/2)^2 * g z" if "Re z > -1" "Re z < 2" for z
proof (cases "z ∈ ℤ")
case True
with that have "z = 0 ∨ z = 1" by (force elim!: Ints_cases)
moreover have "g 0 * g (1/2) = Gamma (1/2)^2 * g 0"
using fraction_not_in_ints[where 'a = complex, of 2 1] by (simp add: g_def power2_eq_square)
moreover have "g (1/2) * g 1 = Gamma (1/2)^2 * g 1"
using fraction_not_in_ints[where 'a = complex, of 2 1]
by (simp add: g_def power2_eq_square Beta_def algebra_simps)
ultimately show ?thesis by force
next
case False
hence z: "z/2 ∉ ℤ" "(z+1)/2 ∉ ℤ" using Ints_diff[of "z+1" 1] by (auto elim!: Ints_cases)
hence z': "z/2 ∉ ℤ⇩≤⇩0" "(z+1)/2 ∉ ℤ⇩≤⇩0" by (auto elim!: nonpos_Ints_cases)
from z have "1-z/2 ∉ ℤ" "1-((z+1)/2) ∉ ℤ"
using Ints_diff[of 1 "1-z/2"] Ints_diff[of 1 "1-((z+1)/2)"] by auto
hence z'': "1-z/2 ∉ ℤ⇩≤⇩0" "1-((z+1)/2) ∉ ℤ⇩≤⇩0" by (auto elim!: nonpos_Ints_cases)
from z have "g (z/2) * g ((z+1)/2) =
(Gamma (z/2) * Gamma ((z+1)/2)) * (Gamma (1-z/2) * Gamma (1-((z+1)/2))) *
(sin (of_real pi * z/2) * sin (of_real pi * (z+1)/2))"
also from z' Gamma_legendre_duplication_aux[of "z/2"]
have "Gamma (z/2) * Gamma ((z+1)/2) = exp ((1-z) * of_real (ln 2)) * Gamma (1/2) * Gamma z"
also from z'' Gamma_legendre_duplication_aux[of "1-(z+1)/2"]
have "Gamma (1-z/2) * Gamma (1-(z+1)/2) =
Gamma (1-z) * Gamma (1/2) * exp (z * of_real (ln 2))"
finally have "g (z/2) * g ((z+1)/2) = Gamma (1/2)^2 * (Gamma z * Gamma (1-z) *
(2 * (sin (of_real pi*z/2) * sin (of_real pi*(z+1)/2))))"
also have "sin (of_real pi*(z+1)/2) = cos (of_real pi*z/2)"
using cos_sin_eq[of "- of_real pi * z/2", symmetric]
also have "2 * (sin (of_real pi*z/2) * cos (of_real pi*z/2)) = sin (of_real pi * z)"
by (subst sin_times_cos) (simp add: field_simps)
also have "Gamma z * Gamma (1 - z) * sin (complex_of_real pi * z) = g z"
using ‹z ∉ ℤ› by (simp add: g_def)
finally show ?thesis .
qed
have g_eq: "g (z/2) * g ((z+1)/2) = Gamma (1/2)^2 * g z" for z
proof -
define r where "r = ⌊Re z / 2⌋"
have "Gamma (1/2)^2 * g z = Gamma (1/2)^2 * g (z - of_int (2*r))" by (simp only: g.minus_of_int)
also have "of_int (2*r) = 2 * of_int r" by simp
also have "Re z - 2 * of_int r > -1" "Re z - 2 * of_int r < 2" unfolding r_def by linarith+
hence "Gamma (1/2)^2 * g (z - 2 * of_int r) =
g ((z - 2 * of_int r)/2) * g ((z - 2 * of_int r + 1)/2)"
unfolding r_def by (intro g_eq[symmetric]) simp_all
also have "(z - 2 * of_int r) / 2 = z/2 - of_int r" by simp
also have "g … = g (z/2)" by (rule g.minus_of_int)
also have "(z - 2 * of_int r + 1) / 2 = (z + 1)/2 - of_int r" by simp
also have "g … = g ((z+1)/2)" by (rule g.minus_of_int)
finally show ?thesis ..
qed

have g_nz [simp]: "g z ≠ 0" for z :: complex
unfolding g_def using Ints_diff[of 1 "1 - z"]
by (auto simp: Gamma_eq_zero_iff sin_eq_0 dest!: nonpos_Ints_Int)

have h_eq: "h z = (h (z/2) + h ((z+1)/2)) / 2" for z
proof -
have "((λt. g (t/2) * g ((t+1)/2)) has_field_derivative
(g (z/2) * g ((z+1)/2)) * ((h (z/2) + h ((z+1)/2)) / 2)) (at z)"
by (auto intro!: derivative_eq_intros g_g'[THEN DERIV_chain2] simp: field_simps)
hence "((λt. Gamma (1/2)^2 * g t) has_field_derivative
Gamma (1/2)^2 * g z * ((h (z/2) + h ((z+1)/2)) / 2)) (at z)"
by (subst (1 2) g_eq[symmetric]) simp
from DERIV_cmult[OF this, of "inverse ((Gamma (1/2))^2)"]
have "(g has_field_derivative (g z * ((h (z/2) + h ((z+1)/2))/2))) (at z)"
using fraction_not_in_ints[where 'a = complex, of 2 1]
by (simp add: divide_simps Gamma_eq_zero_iff not_in_Ints_imp_not_in_nonpos_Ints)
moreover have "(g has_field_derivative (g z * h z)) (at z)"
using g_g'[of z] by (simp add: ac_simps)
ultimately have "g z * h z = g z * ((h (z/2) + h ((z+1)/2))/2)"
by (intro DERIV_unique)
thus "h z = (h (z/2) + h ((z+1)/2)) / 2" by simp
qed

obtain h' where h'_cont: "continuous_on UNIV h'" and
h_h': "⋀z. (h has_field_derivative h' z) (at z)"
unfolding h_def by (erule Gamma_reflection_aux)

have h'_eq: "h' z = (h' (z/2) + h' ((z+1)/2)) / 4" for z
proof -
have "((λt. (h (t/2) + h ((t+1)/2)) / 2) has_field_derivative
((h' (z/2) + h' ((z+1)/2)) / 4)) (at z)"
by (fastforce intro!: derivative_eq_intros h_h'[THEN DERIV_chain2])
hence "(h has_field_derivative ((h' (z/2) + h' ((z+1)/2))/4)) (at z)"
by (subst (asm) h_eq[symmetric])
from h_h' and this show "h' z = (h' (z/2) + h' ((z+1)/2)) / 4" by (rule DERIV_unique)
qed

have h'_zero: "h' z = 0" for z
proof -
define m where "m = max 1 ¦Re z¦"
define B where "B = {t. abs (Re t) ≤ m ∧ abs (Im t) ≤ abs (Im z)}"
have "closed ({t. Re t ≥ -m} ∩ {t. Re t ≤ m} ∩
{t. Im t ≥ -¦Im z¦} ∩ {t. Im t ≤ ¦Im z¦})"
(is "closed ?B") by (intro closed_Int closed_halfspace_Re_ge closed_halfspace_Re_le
closed_halfspace_Im_ge closed_halfspace_Im_le)
also have "?B = B" unfolding B_def by fastforce
finally have "closed B" .
moreover have "bounded B" unfolding bounded_iff
proof (intro ballI exI)
fix t assume t: "t ∈ B"
have "norm t ≤ ¦Re t¦ + ¦Im t¦" by (rule cmod_le)
also from t have "¦Re t¦ ≤ m" unfolding B_def by blast
also from t have "¦Im t¦ ≤ ¦Im z¦" unfolding B_def by blast
finally show "norm t ≤ m + ¦Im z¦" by - simp
qed
ultimately have compact: "compact B" by (subst compact_eq_bounded_closed) blast

define M where "M = (SUP z∈B. norm (h' z))"
have "compact (h' ` B)"
by (intro compact_continuous_image continuous_on_subset[OF h'_cont] compact) blast+
hence bdd: "bdd_above ((λz. norm (h' z)) ` B)"
using bdd_above_norm[of "h' ` B"] by (simp add: image_comp o_def compact_imp_bounded)
have "norm (h' z) ≤ M" unfolding M_def by (intro cSUP_upper bdd) (simp_all add: B_def m_def)
also have "M ≤ M/2"
proof (subst M_def, subst cSUP_le_iff)
have "z ∈ B" unfolding B_def m_def by simp
thus "B ≠ {}" by auto
next
show "∀z∈B. norm (h' z) ≤ M/2"
proof
fix t :: complex assume t: "t ∈ B"
from h'_eq[of t] t have "h' t = (h' (t/2) + h' ((t+1)/2)) / 4" by (simp add: dist_0_norm)
also have "norm … = norm (h' (t/2) + h' ((t+1)/2)) / 4" by simp
also have "norm (h' (t/2) + h' ((t+1)/2)) ≤ norm (h' (t/2)) + norm (h' ((t+1)/2))"
by (rule norm_triangle_ineq)
also from t have "abs (Re ((t + 1)/2)) ≤ m" unfolding m_def B_def by auto
with t have "t/2 ∈ B" "(t+1)/2 ∈ B" unfolding B_def by auto
hence "norm (h' (t/2)) + norm (h' ((t+1)/2)) ≤ M + M" unfolding M_def
by (intro add_mono cSUP_upper bdd) (auto simp: B_def)
also have "(M + M) / 4 = M / 2" by simp
finally show "norm (h' t) ≤ M/2" by - simp_all
qed
qed (insert bdd, auto simp: cball_eq_empty)
hence "M ≤ 0" by simp
finally show "h' z = 0" by simp
qed
have h_h'_2: "(h has_field_derivative 0) (at z)" for z
using h_h'[of z] h'_zero[of z] by simp

have g_real: "g z ∈ ℝ" if "z ∈ ℝ" for z
unfolding g_def using that by (auto intro!: Reals_mult Gamma_complex_real)
have h_real: "h z ∈ ℝ" if "z ∈ ℝ" for z
unfolding h_def using that by (auto intro!: Reals_mult Reals_add Reals_diff Polygamma_Real)
have g_nz: "g z ≠ 0" for z unfolding g_def using Ints_diff[of 1 "1-z"]
by (auto simp: Gamma_eq_zero_iff sin_eq_0)

from h'_zero h_h'_2 have "∃c. ∀z∈UNIV. h z = c"
by (intro has_field_derivative_zero_constant) (simp_all add: dist_0_norm)
then obtain c where c: "⋀z. h z = c" by auto
have "∃u. u ∈ closed_segment 0 1 ∧ Re (g 1) - Re (g 0) = Re (h u * g u * (1 - 0))"
by (intro complex_mvt_line g_g')
then obtain u where u: "u ∈ closed_segment 0 1" "Re (g 1) - Re (g 0) = Re (h u * g u)"
by auto
from u(1) have u': "u ∈ ℝ" unfolding closed_segment_def
by (auto simp: scaleR_conv_of_real)
from u' g_real[of u] g_nz[of u] have "Re (g u) ≠ 0" by (auto elim!: Reals_cases)
with u(2) c[of u] g_real[of u] g_nz[of u] u'
have "Re c = 0" by (simp add: complex_is_Real_iff g.of_1)
with h_real[of 0] c[of 0] have "c = 0" by (auto elim!: Reals_cases)
with c have A: "h z * g z = 0" for z by simp
hence "(g has_field_derivative 0) (at z)" for z using g_g'[of z] by simp
hence "∃c'. ∀z∈UNIV. g z = c'" by (intro has_field_derivative_zero_constant) simp_all
then obtain c' where c: "⋀z. g z = c'" by (force simp: dist_0_norm)
from this[of 0] have "c' = pi" unfolding g_def by simp
with c have "g z = pi" by simp

show ?thesis
proof (cases "z ∈ ℤ")
case False
with ‹g z = pi› show ?thesis by (auto simp: g_def divide_simps)
next
case True
then obtain n where n: "z = of_int n" by (elim Ints_cases)
with sin_eq_0[of "of_real pi * z"] have "sin (of_real pi * z) = 0" by force
moreover have "of_int (1 - n) ∈ ℤ⇩≤⇩0" if "n > 0" using that by (intro nonpos_Ints_of_int) simp
ultimately show ?thesis using n
by (cases "n ≤ 0") (auto simp: Gamma_eq_zero_iff nonpos_Ints_of_int)
qed
qed

lemma rGamma_reflection_complex:
"rGamma z * rGamma (1 - z :: complex) = sin (of_real pi * z) / of_real pi"
using Gamma_reflection_complex[of z]
by (simp add: Gamma_def field_split_simps split: if_split_asm)

lemma rGamma_reflection_complex':
"rGamma z * rGamma (- z :: complex) = -z * sin (of_real pi * z) / of_real pi"
proof -
have "rGamma z * rGamma (-z) = -z * (rGamma z * rGamma (1 - z))"
using rGamma_plus1[of "-z", symmetric] by simp
also have "rGamma z * rGamma (1 - z) = sin (of_real pi * z) / of_real pi"
by (rule rGamma_reflection_complex)
finally show ?thesis by simp
qed

lemma Gamma_reflection_complex':
"Gamma z * Gamma (- z :: complex) = - of_real pi / (z * sin (of_real pi * z))"
using rGamma_reflection_complex'[of z] by (force simp add: Gamma_def field_split_simps mult_ac)

lemma Gamma_one_half_real: "Gamma (1/2 :: real) = sqrt pi"
proof -
from Gamma_reflection_complex[of "1/2"] fraction_not_in_ints[where 'a = complex, of 2 1]
have "Gamma (1/2 :: complex)^2 = of_real pi" by (simp add: power2_eq_square)
hence "of_real pi = Gamma (complex_of_real (1/2))^2" by simp
also have "… = of_real ((Gamma (1/2))^2)" by (subst Gamma_complex_of_real) simp_all
finally have "Gamma (1/2)^2 = pi" by (subst (asm) of_real_eq_iff) simp_all
moreover have "Gamma (1/2 :: real) ≥ 0" using Gamma_real_pos[of "1/2"] by simp
ultimately show ?thesis by (rule real_sqrt_unique [symmetric])
qed

lemma Gamma_one_half_complex: "Gamma (1/2 :: complex) = of_real (sqrt pi)"
proof -
have "Gamma (1/2 :: complex) = Gamma (of_real (1/2))" by simp
also have "… = of_real (sqrt pi)" by (simp only: Gamma_complex_of_real Gamma_one_half_real)
finally show ?thesis .
qed

theorem Gamma_legendre_duplication:
fixes z :: complex
assumes "z ∉ ℤ⇩≤⇩0" "z + 1/2 ∉ ℤ⇩≤⇩0"
shows "Gamma z * Gamma (z + 1/2) =
exp ((1 - 2*z) * of_real (ln 2)) * of_real (sqrt pi) * Gamma (2*z)"
using Gamma_legendre_duplication_aux[OF assms] by (simp add: Gamma_one_half_complex)

end

subsection✐‹tag unimportant› ‹Limits and residues›

text ‹
The inverse of the Gamma function has simple zeros:
›

lemma rGamma_zeros:
"(λz. rGamma z / (z + of_nat n)) ─ (- of_nat n) → ((-1)^n * fact n :: 'a :: Gamma)"
proof (subst tendsto_cong)
let ?f = "λz. pochhammer z n * rGamma (z + of_nat (Suc n)) :: 'a"
from eventually_at_ball'[OF zero_less_one, of "- of_nat n :: 'a" UNIV]
show "eventually (λz. rGamma z / (z + of_nat n) = ?f z) (at (- of_nat n))"
by (subst pochhammer_rGamma[of _ "Suc n"])
(auto elim!: eventually_mono simp: field_split_simps pochhammer_rec' eq_neg_iff_add_eq_0)
have "isCont ?f (- of_nat n)" by (intro continuous_intros)
thus "?f ─ (- of_nat n) → (- 1) ^ n * fact n" unfolding isCont_def
qed

text ‹
The simple zeros of the inverse of the Gamma function correspond to simple poles of the Gamma function,
and their residues can easily be computed from the limit we have just proven:
›

lemma Gamma_poles: "filterlim Gamma at_infinity (at (- of_nat n :: 'a :: Gamma))"
proof -
from eventually_at_ball'[OF zero_less_one, of "- of_nat n :: 'a" UNIV]
have "eventually (λz. rGamma z ≠ (0 :: 'a)) (at (- of_nat n))"
by (auto elim!: eventually_mono nonpos_Ints_cases'
simp: rGamma_eq_zero_iff dist_of_nat dist_minus)
with isCont_rGamma[of "- of_nat n :: 'a", OF continuous_ident]
have "filterlim (λz. inverse (rGamma z) :: 'a) at_infinity (at (- of_nat n))"
unfolding isCont_def by (intro filterlim_compose[OF filterlim_inverse_at_infinity])
moreover have "(λz. inverse (rGamma z) :: 'a) = Gamma"
by (intro ext) (simp add: rGamma_inverse_Gamma)
ultimately show ?thesis by (simp only: )
qed

lemma Gamma_residues:
"(λz. Gamma z * (z + of_nat n)) ─ (- of_nat n) → ((-1)^n / fact n :: 'a :: Gamma)"
proof (subst tendsto_cong)
let ?c = "(- 1) ^ n / fact n :: 'a"
from eventually_at_ball'[OF zero_less_one, of "- of_nat n :: 'a" UNIV]
show "eventually (λz. Gamma z * (z + of_nat n) = inverse (rGamma z / (z + of_nat n)))
(at (- of_nat n))"
by (auto elim!: eventually_mono simp: field_split_simps rGamma_inverse_Gamma)
have "(λz. inverse (rGamma z / (z + of_nat n))) ─ (- of_nat n) →
inverse ((- 1) ^ n * fact n :: 'a)"
by (intro tendsto_intros rGamma_zeros) simp_all
also have "inverse ((- 1) ^ n * fact n) = ?c"
by (simp_all add: field_simps flip: power_mult_distrib)
finally show "(λz. inverse (rGamma z / (z + of_nat n))) ─ (- of_nat n) → ?c" .
qed

subsection ‹Alternative definitions›

subsubsection ‹Variant of the Euler form›

definition Gamma_series_euler' where
"Gamma_series_euler' z n =
inverse z * (∏k=1..n. exp (z * of_real (ln (1 + inverse (of_nat k)))) / (1 + z / of_nat k))"

context
begin
private lemma Gamma_euler'_aux1:
fixes z :: "'a :: {real_normed_field,banach}"
assumes n: "n > 0"
shows "exp (z * of_real (ln (of_nat n + 1))) = (∏k=1..n. exp (z * of_real (ln (1 + 1 / of_nat k))))"
proof -
have "(∏k=1..n. exp (z * of_real (ln (1 + 1 / of_nat k)))) =
exp (z * of_real (∑k = 1..n. ln (1 + 1 / real_of_nat k)))"
by (subst exp_sum [symmetric]) (simp_all add: sum_distrib_left)
also have "(∑k=1..n. ln (1 + 1 / of_nat k) :: real) = ln (∏k=1..n. 1 + 1 / real_of_nat k)"
by (subst ln_prod [symmetric]) (auto intro!: add_pos_nonneg)
also have "(∏k=1..n. 1 + 1 / of_nat k :: real) = (∏k=1..n. (of_nat k + 1) / of_nat k)"
by (intro prod.cong) (simp_all add: field_split_simps)
also have "(∏k=1..n. (of_nat k + 1) / of_nat k :: real) = of_nat n + 1"
by (induction n) (simp_all add: prod.nat_ivl_Suc' field_split_simps)
finally show ?thesis ..
qed

theorem Gamma_series_euler':
assumes z: "(z :: 'a :: Gamma) ∉ ℤ⇩≤⇩0"
shows "(λn. Gamma_series_euler' z n) ⇢ Gamma z"
proof (rule Gamma_seriesI, rule Lim_transform_eventually)
let ?f = "λn. fact n * exp (z * of_real (ln (of_nat n + 1))) / pochhammer z (n + 1)"
let ?r = "λn. ?f n / Gamma_series z n"
let ?r' = "λn. exp (z * of_real (ln (of_nat (Suc n) / of_nat n)))"
from z have z': "z ≠ 0" by auto

have "eventually (λn. ?r' n = ?r n) sequentially"
using z by (auto simp: field_split_simps Gamma_series_def ring_distribs exp_diff ln_div add_ac
intro: eventually_mono eventually_gt_at_top[of "0::nat"] dest: pochhammer_eq_0_imp_nonpos_Int)
moreover have "?r' ⇢ exp (z * of_real (ln 1))"
by (intro tendsto_intros LIMSEQ_Suc_n_over_n) simp_all
ultimately show "?r ⇢ 1" by (force intro: Lim_transform_eventually)

from eventually_gt_at_top[of "0::nat"]
show "eventually (λn. ?r n = Gamma_series_euler' z n / Gamma_series z n) sequentially"
proof eventually_elim
fix n :: nat assume n: "n > 0"
from n z' have "Gamma_series_euler' z n =
exp (z * of_real (ln (of_nat n + 1))) / (z * (∏k=1..n. (1 + z / of_nat k)))"
by (subst Gamma_euler'_aux1)
prod_inversef[symmetric] divide_inverse)
also have "(∏k=1..n. (1 + z / of_nat k)) = pochhammer (z + 1) n / fact n"
proof (cases n)
case (Suc n')
then show ?thesis
unfolding pochhammer_prod fact_prod
by (simp add:  atLeastLessThanSuc_atLeastAtMost field_simps prod_dividef
prod.atLeast_Suc_atMost_Suc_shift del: prod.cl_ivl_Suc)
qed auto
also have "z * … = pochhammer z (Suc n) / fact n" by (simp add: pochhammer_rec)
finally show "?r n = Gamma_series_euler' z n / Gamma_series z n" by simp
qed
qed

end

subsubsection ‹Weierstrass form›

definition Gamma_series_Weierstrass :: "'a :: {banach,real_normed_field} ⇒ nat ⇒ 'a" where
"Gamma_series_Weierstrass z n =
exp (-euler_mascheroni * z) / z * (∏k=1..n. exp (z / of_nat k) / (1 + z / of_nat k))"

definition✐‹tag unimportant›
rGamma_series_Weierstrass :: "'a :: {banach,real_normed_field} ⇒ nat ⇒ 'a" where
"rGamma_series_Weierstrass z n =
exp (euler_mascheroni * z) * z * (∏k=1..n. (1 + z / of_nat k) * exp (-z / of_nat k))"

lemma Gamma_series_Weierstrass_nonpos_Ints:
"eventually (λk. Gamma_series_Weierstrass (- of_nat n) k = 0) sequentially"
using eventually_ge_at_top[of n] by eventually_elim (auto simp: Gamma_series_Weierstrass_def)

lemma rGamma_series_Weierstrass_nonpos_Ints:
"eventually (λk. rGamma_series_Weierstrass (- of_nat n) k = 0) sequentially"
using eventually_ge_at_top[of n] by eventually_elim (auto simp: rGamma_series_Weierstrass_def)

theorem Gamma_Weierstrass_complex: "Gamma_series_Weierstrass z ⇢ Gamma (z :: complex)"
proof (cases "z ∈ ℤ⇩≤⇩0")
case True
then obtain n where "z = - of_nat n" by (elim nonpos_Ints_cases')
also from True have "Gamma_series_Weierstrass … ⇢ Gamma z"
by (simp add: tendsto_cong[OF Gamma_series_Weierstrass_nonpos_Ints] Gamma_nonpos_Int)
finally show ?thesis .
next
case False
hence z: "z ≠ 0" by auto
let ?f = "(λx. ∏x = Suc 0..x. exp (z / of_nat x) / (1 + z / of_nat x))"
have A: "exp (ln (1 + z / of_nat n)) = (1 + z / of_nat n)" if "n ≥ 1" for n :: nat
using False that by (subst exp_Ln) (auto simp: field_simps dest!: plus_of_nat_eq_0_imp)
have "(λn. ∑k=1..n. z / of_nat k - ln (1 + z / of_nat k)) ⇢ ln_Gamma z + euler_mascheroni * z + ln z"
using ln_Gamma_series'_aux[OF False]
by (simp only: atLeastLessThanSuc_atLeastAtMost [symmetric] One_nat_def
sum.shift_bounds_Suc_ivl sums_def atLeast0LessThan)
from tendsto_exp[OF this] False z have "?f ⇢ z * exp (euler_mascheroni * z) * Gamma z"
from tendsto_mult[OF tendsto_const[of "exp (-euler_mascheroni * z) / z"] this] z
show "Gamma_series_Weierstrass z ⇢ Gamma z"
by (simp add: exp_minus field_split_simps Gamma_series_Weierstrass_def [abs_def])
qed

lemma tendsto_complex_of_real_iff: "((λx. complex_of_real (f x)) ⤏ of_real c) F = (f ⤏ c) F"
by (rule tendsto_of_real_iff)

lemma Gamma_Weierstrass_real: "Gamma_series_Weierstrass x ⇢ Gamma (x :: real)"
using Gamma_Weierstrass_complex[of "of_real x"] unfolding Gamma_series_Weierstrass_def[abs_def]
by (subst tendsto_complex_of_real_iff [symmetric])

lemma rGamma_Weierstrass_complex: "rGamma_series_Weierstrass z ⇢ rGamma (z :: complex)"
proof (cases "z ∈ ℤ⇩≤⇩0")
case True
then obtain n where "z = - of_nat n" by (elim nonpos_Ints_cases')
also from True have "rGamma_series_Weierstrass … ⇢ rGamma z"
by (simp add: tendsto_cong[OF rGamma_series_Weierstrass_nonpos_Ints] rGamma_nonpos_Int)
finally show ?thesis .
next
case False
have "rGamma_series_Weierstrass z = (λn. inverse (Gamma_series_Weierstrass z n))"
exp_minus divide_inverse prod_inversef[symmetric] mult_ac)
also from False have "… ⇢ inverse (Gamma z)"
by (intro tendsto_intros Gamma_Weierstrass_complex) (simp add: Gamma_eq_zero_iff)
finally show ?thesis by (simp add: Gamma_def)
qed

subsubsection ‹Binomial coefficient form›

lemma Gamma_gbinomial:
"(λn. ((z + of_nat n) gchoose n) * exp (-z * of_real (ln (of_nat n)))) ⇢ rGamma (z+1)"
proof (cases "z = 0")
case False
show ?thesis
proof (rule Lim_transform_eventually)
let ?powr = "λa b. exp (b * of_real (ln (of_nat a)))"
show "eventually (λn. rGamma_series z n / z =
((z + of_nat n) gchoose n) * ?powr n (-z)) sequentially"
proof (intro always_eventually allI)
fix n :: nat
from False have "((z + of_nat n) gchoose n) = pochhammer z (Suc n) / z / fact n"
also have "pochhammer z (Suc n) / z / fact n * ?powr n (-z) = rGamma_series z n / z"
by (simp add: rGamma_series_def field_split_simps exp_minus)
finally show "rGamma_series z n / z = ((z + of_nat n) gchoose n) * ?powr n (-z)" ..
qed

from False have "(λn. rGamma_series z n / z) ⇢ rGamma z / z" by (intro tendsto_intros)
also from False have "rGamma z / z = rGamma (z + 1)" using rGamma_plus1[of z]
finally show "(λn. rGamma_series z n / z) ⇢ rGamma (z+1)" .
qed

lemma gbinomial_minus': "(a + of_nat b) gchoose b = (- 1) ^ b * (- (a + 1) gchoose b)"
by (subst gbinomial_minus) (simp add: power_mult_distrib [symmetric])

lemma gbinomial_asymptotic:
fixes z :: "'a :: Gamma"
shows "(λn. (z gchoose n) / ((-1)^n / exp ((z+1) * of_real (ln (real n))))) ⇢
inverse (Gamma (- z))"
unfolding rGamma_inverse_Gamma [symmetric] using Gamma_gbinomial[of "-z-1"]
by (subst (asm) gbinomial_minus')

lemma fact_binomial_limit:
"(λn. of_nat ((k + n) choose n) / of_nat (n ^ k) :: 'a :: Gamma) ⇢ 1 / fact k"
proof (rule Lim_transform_eventually)
have "(λn. of_nat ((k + n) choose n) / of_real (exp (of_nat k * ln (real_of_nat n))))
⇢ 1 / Gamma (of_nat (Suc k) :: 'a)" (is "?f ⇢ _")
using Gamma_gbinomial[of "of_nat k :: 'a"]
also have "Gamma (of_nat (Suc k)) = fact k" by (simp add: Gamma_fact)
finally show "?f ⇢ 1 / fact k" .

show "eventually (λn. ?f n = of_nat ((k + n) choose n) / of_nat (n ^ k)) sequentially"
using eventually_gt_at_top[of "0::nat"]
proof eventually_elim
fix n :: nat assume n: "n > 0"
from n have "exp (real_of_nat k * ln (real_of_nat n)) = real_of_nat (n^k)"
thus "?f n = of_nat ((k + n) choose n) / of_nat (n ^ k)" by simp
qed
qed

lemma binomial_asymptotic':
"(λn. of_nat ((k + n) choose n) / (of_nat (n ^ k) / fact k) :: 'a :: Gamma) ⇢ 1"
using tendsto_mult[OF fact_binomial_limit[of k] tendsto_const[of "fact k :: 'a"]] by simp

lemma gbinomial_Beta:
assumes "z + 1 ∉ ℤ⇩≤⇩0"
shows   "((z::'a::Gamma) gchoose n) = inverse ((z + 1) * Beta (z - of_nat n + 1) (of_nat n + 1))"
using assms
proof (induction n arbitrary: z)
case 0
hence "z + 2 ∉ ℤ⇩≤⇩0"
using plus_one_in_nonpos_Ints_imp[of "z+1"] by (auto simp: add.commute)
with 0 show ?case
by (auto simp: Beta_def Gamma_eq_zero_iff Gamma_plus1 [symmetric] add.commute)
next
case (Suc n z)
show ?case
proof (cases "z ∈ ℤ⇩≤⇩0")
case True
with Suc.prems have "z = 0"
by (auto elim!: nonpos_Ints_cases simp: algebra_simps one_plus_of_int_in_nonpos_Ints_iff)
show ?thesis
proof (cases "n = 0")
case True
with ‹z = 0› show ?thesis
by (simp add: Beta_def Gamma_eq_zero_iff Gamma_plus1 [symmetric])
next
case False
with ‹z = 0› show ?thesis
by (simp_all add: Beta_pole1 one_minus_of_nat_in_nonpos_Ints_iff gbinomial_1)
qed
next
case False
have "(z gchoose (Suc n)) = ((z - 1 + 1) gchoose (Suc n))" by simp
also have "… = (z - 1 gchoose n) * ((z - 1) + 1) / of_nat (Suc n)"
by (subst gbinomial_factors) (simp add: field_simps)
also from False have "… = inverse (of_nat (Suc n) * Beta (z - of_nat n) (of_nat (Suc n)))"
(is "_ = inverse ?x") by (subst Suc.IH) (simp_all add: field_simps Beta_pole1)
also have "of_nat (Suc n) ∉ (ℤ⇩≤⇩0 :: 'a set)" by (subst of_nat_in_nonpos_Ints_iff) simp_all
hence "?x = (z + 1) * Beta (z - of_nat (Suc n) + 1) (of_nat (Suc n) + 1)"
by (subst Beta_plus1_right [symmetric]) simp_all
finally show ?thesis .
qed
qed

theorem gbinomial_Gamma:
assumes "z + 1 ∉ ℤ⇩≤⇩0"
shows   "(z gchoose n) = Gamma (z + 1) / (fact n * Gamma (z - of_nat n + 1))"
proof -
have "(z gchoose n) = Gamma (z + 2) / (z + 1) / (fact n * Gamma (z - of_nat n + 1))"
also from assms have "Gamma (z + 2) / (z + 1) = Gamma (z + 1)"
finally show ?thesis .
qed

subsubsection ‹Integral form›

lemma integrable_on_powr_from_0':
assumes a: "a > (-1::real)" and c: "c ≥ 0"
shows   "(λx. x powr a) integrable_on {0<..c}"
proof -
from c have *: "{0<..c} - {0..c} = {}" "{0..c} - {0<..c} = {0}" by auto
show ?thesis
by (rule integrable_spike_set [OF integrable_on_powr_from_0[OF a c]]) (simp_all add: *)
qed

lemma absolutely_integrable_Gamma_integral:
assumes "Re z > 0" "a > 0"
shows   "(λt. complex_of_real t powr (z - 1) / of_real (exp (a * t)))
absolutely_integrable_on {0<..}" (is "?f absolutely_integrable_on _")
proof -
have "((λx. (Re z - 1) * (ln x / x)) ⤏ (Re z - 1) * 0) at_top"
by (intro tendsto_intros ln_x_over_x_tendsto_0)
hence "((λx. ((Re z - 1) * ln x) / x) ⤏ 0) at_top" by simp
from order_tendstoD(2)[OF this, of "a/2"] and ‹a > 0›
have "eventually (λx. (Re z - 1) * ln x / x < a/2) at_top" by simp
from eventually_conj[OF this eventually_gt_at_top[of 0]]
obtain x0 where "∀x≥x0. (Re z - 1) * ln x / x < a/2 ∧ x > 0"
by (auto simp: eventually_at_top_linorder)
hence "x0 > 0" by simp
have "x powr (Re z - 1) / exp (a * x) < exp (-(a/2) * x)" if "x ≥ x0" for x
proof -
from that and ‹∀x≥x0. _› have x: "(Re z - 1) * ln x / x < a / 2" "x > 0" by auto
have "x powr (Re z - 1) = exp ((Re z - 1) * ln x)"
using ‹x > 0› by (simp add: powr_def)
also from x have "(Re z - 1) * ln x < (a * x) / 2" by (simp add: field_simps)
qed
note x0 = ‹x0 > 0› this

have "?f absolutely_integrable_on ({0<..x0} ∪ {x0..})"
proof (rule set_integrable_Un)
show "?f absolutely_integrable_on {0<..x0}"
unfolding set_integrable_def
proof (rule Bochner_Integration.integrable_bound [OF _ _ AE_I2])
show "integrable lebesgue (λx. indicat_real {0<..x0} x *⇩R x powr (Re z - 1))"
using x0(1) assms
by (intro nonnegative_absolutely_integrable_1 [unfolded set_integrable_def] integrable_on_powr_from_0') auto
show "(λx. indicat_real {0<..x0} x *⇩R (x powr (z - 1) / exp (a * x))) ∈ borel_measurable lebesgue"
by (intro measurable_completion)
(auto intro!: borel_measurable_continuous_on_indicator continuous_intros)
fix x :: real
have "x powr (Re z - 1) / exp (a * x) ≤ x powr (Re z - 1) / 1" if "x ≥ 0"
using that assms by (intro divide_left_mono) auto
thus "norm (indicator {0<..x0} x *⇩R ?f x) ≤
norm (indicator {0<..x0} x *⇩R x powr (Re z - 1))"
by (simp_all add: norm_divide norm_powr_real_powr indicator_def)
qed
next
show "?f absolutely_integrable_on {x0..}"
unfolding set_integrable_def
proof (rule Bochner_Integration.integrable_bound [OF _ _ AE_I2])
show "integrable lebesgue (λx. indicat_real {x0..} x *⇩R exp (- (a / 2) * x))" using assms
by (intro nonnegative_absolutely_integrable_1 [unfolded set_integrable_def] integrable_on_exp_minus_to_infinity) auto
show "(λx. indicat_real {x0..} x *⇩R (x powr (z - 1) / exp (a * x))) ∈ borel_measurable lebesgue" using x0(1)
by (intro measurable_completion)
(auto intro!: borel_measurable_continuous_on_indicator continuous_intros)
fix x :: real
show "norm (indicator {x0..} x *⇩R ?f x) ≤
norm (indicator {x0..} x *⇩R exp (-(a/2) * x))" using x0
by (auto simp: norm_divide norm_powr_real_powr indicator_def less_imp_le)
qed
qed auto
also have "{0<..x0} ∪ {x0..} = {0<..}" using x0(1) by auto
finally show ?thesis .
qed

lemma integrable_Gamma_integral_bound:
fixes a c :: real
assumes a: "a > -1" and c: "c ≥ 0"
defines "f ≡ λx. if x ∈ {0..c} then x powr a else exp (-x/2)"
shows   "f integrable_on {0..}"
proof -
have "f integrable_on {0..c}"
by (rule integrable_spike_finite[of "{}", OF _ _ integrable_on_powr_from_0[of a c]])
(insert a c, simp_all add: f_def)
moreover have A: "(λx. exp (-x/2)) integrable_on {c..}"
using integrable_on_exp_minus_to_infinity[of "1/2"] by simp
have "f integrable_on {c..}"
by (rule integrable_spike_finite[of "{c}", OF _ _ A]) (simp_all add: f_def)
ultimately show "f integrable_on {0..}"
by (rule integrable_Un') (insert c, auto simp: max_def)
qed

theorem Gamma_integral_complex:
assumes z: "Re z > 0"
shows   "((λt. of_real t powr (z - 1) / of_real (exp t)) has_integral Gamma z) {0..}"
proof -
have A: "((λt. (of_real t) powr (z - 1) * of_real ((1 - t) ^ n))
has_integral (fact n / pochhammer z (n+1))) {0..1}"
if "Re z > 0" for n z using that
proof (induction n arbitrary: z)
case 0
have "((λt. complex_of_real t powr (z - 1)) has_integral
(of_real 1 powr z / z - of_real 0 powr z / z)) {0..1}" using 0
by (intro fundamental_theorem_of_calculus_interior)
(auto intro!: continuous_intros derivative_eq_intros has_vector_derivative_real_field)
thus ?case by simp
next
case (Suc n)
let ?f = "λt. complex_of_real t powr z / z"
let ?f' = "λt. complex_of_real t powr (z - 1)"
let ?g = "λt. (1 - complex_of_real t) ^ Suc n"
let ?g' = "λt. - ((1 - complex_of_real t) ^ n) * of_nat (Suc n)"
have "((λt. ?f' t * ?g t) has_integral
(of_nat (Suc n)) * fact n / pochhammer z (n+2)) {0..1}"
(is "(_ has_integral ?I) _")
proof (rule integration_by_parts_interior[where f' = ?f' and g = ?g])
from Suc.prems show "continuous_on {0..1} ?f" "continuous_on {0..1} ?g"
by (auto intro!: continuous_intros)
next
fix t :: real assume t: "t ∈ {0<..<1}"
show "(?f has_vector_derivative ?f' t) (at t)" using t Suc.prems
by (auto intro!: derivative_eq_intros has_vector_derivative_real_field)
show "(?g has_vector_derivative ?g' t) (at t)"
by (rule has_vector_derivative_real_field derivative_eq_intros refl)+ simp_all
next
from Suc.prems have [simp]: "z ≠ 0" by auto
from Suc.prems have A: "Re (z + of_nat n) > 0" for n by simp
have [simp]: "z + of_nat n ≠ 0" "z + 1 + of_nat n ≠ 0" for n
using A[of n] A[of "Suc n"] by (auto simp add: add.assoc simp del: plus_complex.sel)
have "((λx. of_real x powr z * of_real ((1 - x) ^ n) * (- of_nat (Suc n) / z)) has_integral
fact n / pochhammer (z+1) (n+1) * (- of_nat (Suc n) / z)) {0..1}"
(is "(?A has_integral ?B) _")
also have "?A = (λt. ?f t * ?g' t)" by (intro ext) (simp_all add: field_simps)
also have "?B = - (of_nat (Suc n) * fact n / pochhammer z (n+2))"
prod.shift_bounds_cl_Suc_ivl del: of_nat_Suc)
finally show "((λt. ?f t * ?g' t) has_integral (?f 1 * ?g 1 - ?f 0 * ?g 0 - ?I)) {0..1}"
by simp
thus ?case by simp
qed

have B: "((λt. if t ∈ {0..of_nat n} then
of_real t powr (z - 1) * (1 - of_real t / of_nat n) ^ n else 0)
has_integral (of_nat n powr z * fact n / pochhammer z (n+1))) {0..}" for n
proof (cases "n > 0")
case [simp]: True
hence [simp]: "n ≠ 0" by auto
with has_integral_affinity01[OF A[OF z, of n], of "inverse (of_nat n)" 0]
have "((λx. (of_nat n - of_real x) ^ n * (of_real x / of_nat n) powr (z - 1) / of_nat n ^ n)
has_integral fact n * of_nat n / pochhammer z (n+1)) ((λx. real n * x)`{0..1})"
(is "(?f has_integral ?I) ?ivl") by (simp add: field_simps scaleR_conv_of_real)
also from True have "((λx. real n*x)`{0..1}) = {0..real n}"
by (subst image_mult_atLeastAtMost) simp_all
also have "?f = (λx. (of_real x / of_nat n) powr (z - 1) * (1 - of_real x / of_nat n) ^ n)"
using True by (intro ext) (simp add: field_simps)
finally have "((λx. (of_real x / of_nat n) powr (z - 1) * (1 - of_real x / of_nat n) ^ n)
has_integral ?I) {0..real n}" (is ?P) .
also have "?P ⟷ ((λx. exp ((z - 1) * of_real (ln (x / of_nat n))) * (1 - of_real x / of_nat n) ^ n)
has_integral ?I) {0..real n}"
by (intro has_integral_spike_finite_eq[of "{0}"]) (auto simp: powr_def Ln_of_real [symmetric])
also have "… ⟷ ((λx. exp ((z - 1) * of_real (ln x - ln (of_nat n))) * (1 - of_real x / of_nat n) ^ n)
has_integral ?I) {0..real n}"
by (intro has_integral_spike_finite_eq[of "{0}"]) (simp_all add: ln_div)
finally have … .
note B = has_integral_mult_right[OF this, of "exp ((z - 1) * ln (of_nat n))"]
have "((λx. exp ((z - 1) * of_real (ln x)) * (1 - of_real x / of_nat n) ^ n)
has_integral (?I * exp ((z - 1) * ln (of_nat n)))) {0..real n}" (is ?P)
by (insert B, subst (asm) mult.assoc [symmetric], subst (asm) exp_add [symmetric])
also have "?P ⟷ ((λx. of_real x powr (z - 1) * (1 - of_real x / of_nat n) ^ n)
has_integral (?I * exp ((z - 1) * ln (of_nat n)))) {0..real n}"
by (intro has_integral_spike_finite_eq[of "{0}"]) (simp_all add: powr_def Ln_of_real)
also have "fact n * of_nat n / pochhammer z (n+1) * exp ((z - 1) * Ln (of_nat n)) =
(of_nat n powr z * fact n / pochhammer z (n+1))"
by (auto simp add: powr_def algebra_simps exp_diff exp_of_real)
finally show ?thesis by (subst has_integral_restrict) simp_all
next
case False
thus ?thesis by (subst has_integral_restrict) (simp_all add: has_integral_refl)
qed

have "eventually (λn. Gamma_series z n =
of_nat n powr z * fact n / pochhammer z (n+1)) sequentially"
using eventually_gt_at_top[of "0::nat"]
by eventually_elim (simp add: powr_def algebra_simps Ln_of_nat Gamma_series_def)
from this and Gamma_series_LIMSEQ[of z]
have C: "(λk. of_nat k powr z * fact k / pochhammer z (k+1)) ⇢ Gamma z"
by (blast intro: Lim_transform_eventually)
{
fix x :: real assume x: "x ≥ 0"
have lim_exp: "(λk. (1 - x / real k) ^ k) ⇢ exp (-x)"
using tendsto_exp_limit_sequentially[of "-x"] by simp
have "(λk. of_real x powr (z - 1) * of_real ((1 - x / of_nat k) ^ k))
⇢ of_real x powr (z - 1) * of_real (exp (-x))" (is ?P)
by (intro tendsto_intros lim_exp)
also from eventually_gt_at_top[of "nat ⌈x⌉"]
have "eventually (λk. of_nat k > x) sequentially" by eventually_elim linarith
hence "?P ⟷ (λk. if x ≤ of_nat k then
of_real x powr (z - 1) * of_real ((1 - x / of_nat k) ^ k) else 0)
⇢ of_real x powr (z - 1) * of_real (exp (-x))"
by (intro tendsto_cong) (auto elim!: eventually_mono)
finally have … .
}
hence D: "∀x∈{0..}. (λk. if x ∈ {0..real k} then
of_real x powr (z - 1) * (1 - of_real x / of_nat k) ^ k else 0)
⇢ of_real x powr (z - 1) / of_real (exp x)"
by (simp add: exp_minus field_simps cong: if_cong)

have "((λx. (Re z - 1) * (ln x / x)) ⤏ (Re z - 1) * 0) at_top"
by (intro tendsto_intros ln_x_over_x_tendsto_0)
hence "((λx. ((Re z - 1) * ln x) / x) ⤏ 0) at_top" by simp
from order_tendstoD(2)[OF this, of "1/2"]
have "eventually (λx. (Re z - 1) * ln x / x < 1/2) at_top" by simp
from eventually_conj[OF this eventually_gt_at_top[of 0]]
obtain x0 where "∀x≥x0. (Re z - 1) * ln x / x < 1/2 ∧ x > 0"
by (auto simp: eventually_at_top_linorder)
hence x0: "x0 > 0" "⋀x. x ≥ x0 ⟹ (Re z - 1) * ln x < x / 2" by auto

define h where "h = (λx. if x ∈ {0..x0} then x powr (Re z - 1) else exp (-x/2))"
have le_h: "x powr (Re z - 1) * exp (-x) ≤ h x" if x: "x ≥ 0" for x
proof (cases "x > x0")
case True
from True x0(1) have "x powr (Re z - 1) * exp (-x) = exp ((Re z - 1) * ln x - x)"
also from x0(2)[of x] True have "… < exp (-x/2)"
finally show ?thesis using True by (auto simp add: h_def)
next
case False
from x have "x powr (Re z - 1) * exp (- x) ≤ x powr (Re z - 1) * 1"
by (intro mult_left_mono) simp_all
with False show ?thesis by (auto simp add: h_def)
qed

have E: "∀x∈{0..}. cmod (if x ∈ {0..real k} then of_real x powr (z - 1) *
(1 - complex_of_real x / of_nat k) ^ k else 0) ≤ h x"
(is "∀x∈_. ?f x ≤ _") for k
proof safe
fix x :: real assume x: "x ≥ 0"
{
fix x :: real and n :: nat assume x: "x ≤ of_nat n"
have "(1 - complex_of_real x / of_nat n) = complex_of_real ((1 - x / of_nat n))" by simp
also have "norm … = ¦(1 - x / real n)¦" by (subst norm_of_real) (rule refl)
also from x have "… = (1 - x / real n)" by (intro abs_of_nonneg) (simp_all add: field_split_simps)
finally have "cmod (1 - complex_of_real x / of_nat n) = 1 - x / real n" .
} note D = this
from D[of x k] x
have "?f x ≤ (if of_nat k ≥ x ∧ k > 0 then x powr (Re z - 1) * (1 - x / real k) ^ k else 0)"
by (auto simp: norm_mult norm_powr_real_powr norm_power intro!: mult_nonneg_nonneg)
also have "… ≤ x powr (Re z - 1) * exp  (-x)"
by (auto intro!: mult_left_mono exp_ge_one_minus_x_over_n_power_n)
also from x have "… ≤ h x" by (rule le_h)
finally show "?f x ≤ h x" .
qed

have F: "h integrable_on {0..}" unfolding h_def
by (rule integrable_Gamma_integral_bound) (insert assms x0(1), simp_all)
show ?thesis
by (rule has_integral_dominated_convergence[OF B F E D C])
qed

lemma Gamma_integral_real:
assumes x: "x > (0 :: real)"
shows   "((λt. t powr (x - 1) / exp t) has_integral Gamma x) {0..}"
proof -
have A: "((λt. complex_of_real t powr (complex_of_real x - 1) /
complex_of_real (exp t)) has_integral complex_of_real (Gamma x)) {0..}"
using Gamma_integral_complex[of x] assms by (simp_all add: Gamma_complex_of_real powr_of_real)
have "((λt. complex_of_real (t powr (x - 1) / exp t)) has_integral of_real (Gamma x)) {0..}"
by (rule has_integral_eq[OF _ A]) (simp_all add: powr_of_real [symmetric])
from has_integral_linear[OF this bounded_linear_Re] show ?thesis by (simp add: o_def)
qed

lemma absolutely_integrable_Gamma_integral':
assumes "Re z > 0"
shows   "(λt. complex_of_real t powr (z - 1) / of_real (exp t)) absolutely_integrable_on {0<..}"
using absolutely_integrable_Gamma_integral [OF assms zero_less_one] by simp

lemma Gamma_integral_complex':
assumes z: "Re z > 0"
shows   "((λt. of_real t powr (z - 1) / of_real (exp t)) has_integral Gamma z) {0<..}"
proof -
have "((λt. of_real t powr (z - 1) / of_real (exp t)) has_integral Gamma z) {0..}"
by (rule Gamma_integral_complex) fact+
hence "((λt. if t ∈ {0<..} then of_real t powr (z - 1) / of_real (exp t) else 0)
has_integral Gamma z) {0..}"
by (rule has_integral_spike [of "{0}", rotated 2]) auto
also have "?this = ?thesis"
by (subst has_integral_restrict) auto
finally show ?thesis .
qed

lemma Gamma_conv_nn_integral_real:
assumes "s > (0::real)"
shows   "Gamma s = nn_integral lborel (λt. ennreal (indicator {0..} t * t powr (s - 1) / exp t))"
using nn_integral_has_integral_lebesgue[OF _ Gamma_integral_real[OF assms]] by simp

lemma integrable_Beta:
assumes "a > 0" "b > (0::real)"
shows   "set_integrable lborel {0..1} (λt. t powr (a - 1) * (1 - t) powr (b - 1))"
proof -
define C where "C = max 1 ((1/2) powr (b - 1))"
define D where "D = max 1 ((1/2) powr (a - 1))"
have C: "(1 - x) powr (b - 1) ≤ C" if "x ∈ {0..1/2}" for x
proof (cases "b < 1")
case False
with that have "(1 - x) powr (b - 1) ≤ (1 powr (b - 1))" by (intro powr_mono2) auto
thus ?thesis by (auto simp: C_def)
qed (insert that, auto simp: max.coboundedI1 max.coboundedI2 powr_mono2' powr_mono2 C_def)
have D: "x powr (a - 1) ≤ D" if "x ∈ {1/2..1}" for x
proof (cases "a < 1")
case False
with that have "x powr (a - 1) ≤ (1 powr (a - 1))" by (intro powr_mono2) auto
thus ?thesis by (auto simp: D_def)
next
case True
qed (insert that, auto simp: max.coboundedI1 max.coboundedI2 powr_mono2' powr_mono2 D_def)
have [simp]: "C ≥ 0" "D ≥ 0" by (simp_all add: C_def D_def)

have I1: "set_integrable lborel {0..1/2} (λt. t powr (a - 1) * (1 - t) powr (b - 1))"
unfolding set_integrable_def
proof (rule Bochner_Integration.integrable_bound[OF _ _ AE_I2])
have "(λt. t powr (a - 1)) integrable_on {0..1/2}"
by (rule integrable_on_powr_from_0) (use assms in auto)
hence "(λt. t powr (a - 1)) absolutely_integrable_on {0..1/2}"
by (subst absolutely_integrable_on_iff_nonneg) auto
from integrable_mult_right[OF this [unfolded set_integrable_def], of C]
show "integrable lborel (λx. indicat_real {0..1/2} x *⇩R (C * x powr (a - 1)))"
by (subst (asm) integrable_completion) (auto simp: mult_ac)
next
fix x :: real
have "x powr (a - 1) * (1 - x) powr (b - 1) ≤ x powr (a - 1) * C" if "x ∈ {0..1/2}"
using that by (intro mult_left_mono powr_mono2 C) auto
thus "norm (indicator {0..1/2} x *⇩R (x powr (a - 1) * (1 - x) powr (b - 1))) ≤
norm (indicator {0..1/2} x *⇩R (C * x powr (a - 1)))"
by (auto simp: indicator_def abs_mult mult_ac)
qed (auto intro!: AE_I2 simp: indicator_def)

have I2: "set_integrable lborel {1/2..1} (λt. t powr (a - 1) * (1 - t) powr (b - 1))"
unfolding set_integrable_def
proof (rule Bochner_Integration.integrable_bound[OF _ _ AE_I2])
have "(λt. t powr (b - 1)) integrable_on {0..1/2}"
by (rule integrable_on_powr_from_0) (use assms in auto)
hence "(λt. t powr (b - 1)) integrable_on (cbox 0 (1/2))" by simp
from integrable_affinity[OF this, of "-1" 1]
have "(λt. (1 - t) powr (b - 1)) integrable_on {1/2..1}" by simp
hence "(λt. (1 - t) powr (b - 1)) absolutely_integrable_on {1/2..1}"
by (subst absolutely_integrable_on_iff_nonneg) auto
from integrable_mult_right[OF this [unfolded set_integrable_def], of D]
show "integrable lborel (λx. indicat_real {1/2..1} x *⇩R (D * (1 - x) powr (b - 1)))"
by (subst (asm) integrable_completion) (auto simp: mult_ac)
next
fix x :: real
have "x powr (a - 1) * (1 - x) powr (b - 1) ≤ D * (1 - x) powr (b - 1)" if "x ∈ {1/2..1}"
using that by (intro mult_right_mono powr_mono2 D) auto
thus "norm (indicator {1/2..1} x *⇩R (x powr (a - 1) * (1 - x) powr (b - 1))) ≤
norm (indicator {1/2..1} x *⇩R (D * (1 - x) powr (b - 1)))"
by (auto simp: indicator_def abs_mult mult_ac)
qed (auto intro!: AE_I2 simp: indicator_def)

have "set_integrable lborel ({0..1/2} ∪ {1/2..1}) (λt. t powr (a - 1) * (1 - t) powr (b - 1))"
by (intro set_integrable_Un I1 I2) auto
also have "{0..1/2} ∪ {1/2..1} = {0..(1::real)}" by auto
finally show ?thesis .
qed

lemma integrable_Beta':
assumes "a > 0" "b > (0::real)"
shows   "(λt. t powr (a - 1) * (1 - t) powr (b - 1)) integrable_on {0..1}"
using integrable_Beta[OF assms] by (rule set_borel_integral_eq_integral)

theorem has_integral_Beta_real:
assumes a: "a > 0" and b: "b > (0 :: real)"
shows "((λt. t powr (a - 1) * (1 - t) powr (b - 1)) has_integral Beta a b) {0..1}"
proof -
define B where "B = integral {0..1} (λx. x powr (a - 1) * (1 - x) powr (b - 1))"
have [simp]: "B ≥ 0" unfolding B_def using a b
by (intro integral_nonneg integrable_Beta') auto
from a b have "ennreal (Gamma a * Gamma b) =
(∫⇧+ t. ennreal (indicator {0..} t * t powr (a - 1) / exp t) ∂lborel) *
(∫⇧+ t. ennreal (indicator {0..} t * t powr (b - 1) / exp t) ∂lborel)"
by (subst ennreal_mult') (simp_all add: Gamma_conv_nn_integral_real)
also have "… = (∫⇧+t. ∫⇧+u. ennreal (indicator {0..} t * t powr (a - 1) / exp t) *
ennreal (indicator {0..} u * u powr (b - 1) / exp u) ∂lborel ∂lborel)"
also have "… = (∫⇧+t. ∫⇧+u. ennreal (indicator ({0..}×{0..}) (t,u) * t powr (a - 1) * u powr (b - 1)
/ exp (t + u)) ∂lborel ∂lborel)"
by (intro nn_integral_cong)
(auto simp: indicator_def divide_ennreal ennreal_mult' [symmetric] exp_add)
also have "… = (∫⇧+t. ∫⇧+u. ennreal (indicator ({0..}×{t..}) (t,u) * t powr (a - 1) *
(u - t) powr (b - 1) / exp u) ∂lborel ∂lborel)"
proof (rule nn_integral_cong, goal_cases)
case (1 t)
have "(∫⇧+u. ennreal (indicator ({0..}×{0..}) (t,u) * t powr (a - 1) *
u powr (b - 1) / exp (t + u)) ∂distr lborel borel ((+) (-t))) =
(∫⇧+u. ennreal (indicator ({0..}×{t..}) (t,u) * t powr (a - 1) *
(u - t) powr (b - 1) / exp u) ∂lborel)"
by (subst nn_integral_distr) (auto intro!: nn_integral_cong simp: indicator_def)
thus ?case by (subst (asm) lborel_distr_plus)
qed
also have "… = (∫⇧+u. ∫⇧+t. ennreal (indicator ({0..}×{t..}) (t,u) * t powr (a - 1) *
(u - t) powr (b - 1) / exp u) ∂lborel ∂lborel)"
by (subst lborel_pair.Fubini')
(auto simp: case_prod_unfold indicator_def cong: measurable_cong_sets)
also have "… = (∫⇧+u. ∫⇧+t. ennreal (indicator {0..u} t * t powr (a - 1) * (u - t) powr (b - 1)) *
ennreal (indicator {0..} u / exp u) ∂lborel ∂lborel)"
by (intro nn_integral_cong) (auto simp: indicator_def ennreal_mult' [symmetric])
also have "… = (∫⇧+u. (∫⇧+t. ennreal (indicator {0..u} t * t powr (a - 1) * (u - t) powr (b - 1))
∂lborel) * ennreal (indicator {0..} u / exp u) ∂lborel)"
by (subst nn_integral_multc [symmetric]) auto
also have "… = (∫⇧+u. (∫⇧+t. ennreal (indicator {0..u} t * t powr (a - 1) * (u - t) powr (b - 1))
∂lborel) * ennreal (indicator {0<..} u / exp u) ∂lborel)"
by (intro nn_integral_cong_AE eventually_mono[OF AE_lborel_singleton[of 0]])
(auto simp: indicator_def)
also have "… = (∫⇧+u. ennreal B * ennreal (indicator {0..} u / exp u * u powr (a + b - 1)) ∂lborel)"
proof (intro nn_integral_cong, goal_cases)
case (1 u)
show ?case
proof (cases "u > 0")
case True
have "(∫⇧+t. ennreal (indicator {0..u} t * t powr (a - 1) * (u - t) powr (b - 1)) ∂lborel) =
(∫⇧+t. ennreal (indicator {0..1} t * (u * t) powr (a - 1) * (u - u * t) powr (b - 1))
∂distr lborel borel ((*) (1 / u)))" (is "_ = nn_integral _ ?f")
using True
by (subst nn_integral_distr) (auto simp: indicator_def field_simps intro!: nn_integral_cong)
also have "distr lborel borel ((*) (1 / u)) = density lborel (λ_. u)"
using ‹u > 0› by (subst lborel_distr_mult) auto
also have "nn_integral … ?f = (∫⇧+x. ennreal (indicator {0..1} x * (u * (u * x) powr (a - 1) *
(u * (1 - x)) powr (b - 1))) ∂lborel)" using ‹u > 0›
by (subst nn_integral_density) (auto simp: ennreal_mult' [symmetric] algebra_simps)
also have "… = (∫⇧+x. ennreal (u powr (a + b - 1)) *
ennreal (indicator {0..1} x * x powr (a - 1) *
(1 - x) powr (b - 1)) ∂lborel)" using ‹u > 0› a b
by (intro nn_integral_cong)
(auto simp: indicator_def powr_mult powr_add powr_diff mult_ac ennreal_mult' [symmetric])
also have "… = ennreal (u powr (a + b - 1)) *
(∫⇧+x. ennreal (indicator {0..1} x * x powr (a - 1) *
(1 - x) powr (b - 1)) ∂lborel)"
by (subst nn_integral_cmult) auto
also have "((λx. x powr (a - 1) * (1 - x) powr (b - 1)) has_integral
integral {0..1} (λx. x powr (a - 1) * (1 - x) powr (b - 1))) {0..1}"
using a b by (intro integrable_integral integrable_Beta')
from nn_integral_has_integral_lebesgue[OF _ this] a b
have "(∫⇧+x. ennreal (indicator {0..1} x * x powr (a - 1) *
(1 - x) powr (b - 1)) ∂lborel) = B" by (simp add: mult_ac B_def)
finally show ?thesis using ‹u > 0› by (simp add: ennreal_mult' [symmetric] mult_ac)
qed auto
qed
also have "… = ennreal B * ennreal (Gamma (a + b))"
using a b by (subst nn_integral_cmult) (auto simp: Gamma_conv_nn_integral_real)
also have "… = ennreal (B * Gamma (a + b))"
by (subst (1 2) mult.commute, intro ennreal_mult' [symmetric]) (use a b in auto)
finally have "B = Beta a b" using a b Gamma_real_pos[of "a + b"]
by (subst (asm) ennreal_inj) (auto simp: field_simps Beta_def Gamma_eq_zero_iff)
moreover have "(λt. t powr (a - 1) * (1 - t) powr (b - 1)) integrable_on {0..1}"
by (intro integrable_Beta' a b)
ultimately show ?thesis by (simp add: has_integral_iff B_def)
qed

subsection ‹The Weierstraß product formula for the sine›

theorem sin_product_formula_complex:
fixes z :: complex
shows "(λn. of_real pi * z * (∏k=1..n. 1 - z^2 / of_nat k^2)) ⇢ sin (of_real pi * z)"
proof -
let ?f = "rGamma_series_Weierstrass"
have "(λn. (- of_real pi * inverse z) * (?f z n * ?f (- z) n))
⇢ (- of_real pi * inverse z) * (rGamma z * rGamma (- z))"
by (intro tendsto_intros rGamma_Weierstrass_complex)
also have "(λn. (- of_real pi * inverse z) * (?f z n * ?f (-z) n)) =
(λn. of_real pi * z * (∏k=1..n. 1 - z^2 / of_nat k ^ 2))"
proof
fix n :: nat
have "(- of_real pi * inverse z) * (?f z n * ?f (-z) n) =
of_real pi * z * (∏k=1..n. (of_nat k - z) * (of_nat k + z) / of_nat k ^ 2)"
by (simp add: rGamma_series_Weierstrass_def mult_ac exp_minus
divide_simps prod.distrib[symmetric] power2_eq_square)
also have "(∏k=1..n. (of_nat k - z) * (of_nat k + z) / of_nat k ^ 2) =
(∏k=1..n. 1 - z^2 / of_nat k ^ 2)"
by (intro prod.cong) (simp_all add: power2_eq_square field_simps)
finally show "(- of_real pi * inverse z) * (?f z n * ?f (-z) n) = of_real pi * z * …"
qed
also have "(- of_real pi * inverse z) * (rGamma z * rGamma (- z)) = sin (of_real pi * z)"
by (subst rGamma_reflection_complex') (simp add: field_split_simps)
finally show ?thesis .
qed

lemma sin_product_formula_real:
"(λn. pi * (x::real) * (∏k=1..n. 1 - x^2 / of_nat k^2)) ⇢ sin (pi * x)"
proof -
from sin_product_formula_complex[of "of_real x"]
have "(λn. of_real pi * of_real x * (∏k=1..n. 1 - (of_real x)^2 / (of_nat k)^2))
⇢ sin (of_real pi * of_real x :: complex)" (is "?f ⇢ ?y") .
also have "?f = (λn. of_real (pi * x * (∏k=1..n. 1 - x^2 / (of_nat k^2))))" by simp
also have "?y = of_real (sin (pi * x))" by (simp only: sin_of_real [symmetric] of_real_mult)
finally show ?thesis by (subst (asm) tendsto_of_real_iff)
qed

lemma sin_product_formula_real':
assumes "x ≠ (0::real)"
shows   "(λn. (∏k=1..n. 1 - x^2 / of_nat k^2)) ⇢ sin (pi * x) / (pi * x)"
using tendsto_divide[OF sin_product_formula_real[of x] tendsto_const[of "pi * x"]] assms
by simp

theorem wallis: "(λn. ∏k=1..n. (4*real k^2) / (4*real k^2 - 1)) ⇢ pi / 2"
proof -
from tendsto_inverse[OF tendsto_mult[OF
sin_product_formula_real[of "1/2"] tendsto_const[of "2/pi"]]]
have "(λn. (∏k=1..n. inverse (1 - (1/2)⇧2 / (real k)⇧2))) ⇢ pi/2"
also have "(λn. (∏k=1..n. inverse (1 - (1/2)⇧2 / (real k)⇧2))) =
(λn. (∏k=1..n. (4*real k^2)/(4*real k^2 - 1)))"
by (intro ext prod.cong refl) (simp add: field_split_simps)
finally show ?thesis .
qed

subsection ‹The Solution to the Basel problem›

theorem inverse_squares_sums: "(λn. 1 / (n + 1)⇧2) sums (pi⇧2 / 6)"
proof -
define P where "P x n = (∏k=1..n. 1 - x^2 / of_nat k^2)" for x :: real and n
define K where "K = (∑n. inverse (real_of_nat (Suc n))^2)"
define f where [abs_def]: "f x = (∑n. P x n / of_nat (Suc n)^2)" for x
define g where [abs_def]: "g x = (1 - sin (pi * x) / (pi * x))" for x

have sums: "(λn. P x n / of_nat (Suc n)^2) sums (if x = 0 then K else g x / x^2)" for x
proof (cases "x = 0")
assume x: "x = 0"
have "summable (λn. inverse ((real_of_nat (Suc n))⇧2))"
using inverse_power_summable[of 2] by (subst summable_Suc_iff) simp
thus ?thesis by (simp add: x g_def P_def K_def inverse_eq_divide power_divide summable_sums)
next
assume x: "x ≠ 0"
have "(λn. P x n - P x (Suc n)) sums (P x 0 - sin (pi * x) / (pi * x))"
unfolding P_def using x by (intro telescope_sums' sin_product_formula_real')
also have "(λn. P x n - P x (Suc n)) = (λn. (x^2 / of_nat (Suc n)^2) * P x n)"
unfolding P_def by (simp add: prod.nat_ivl_Suc' algebra_simps)
also have "P x 0 = 1" by (simp add: P_def)
finally have "(λn. x⇧2 / (of_nat (Suc n))⇧2 * P x n) sums (1 - sin (pi * x) / (pi * x))" .
from sums_divide[OF this, of "x^2"] x show ?thesis unfolding g_def by simp
qed

have "continuous_on (ball 0 1) f"
proof (rule uniform_limit_theorem; (intro always_eventually allI)?)
show "uniform_limit (ball 0 1) (λn x. ∑k<n. P x k / of_nat (Suc k)^2) f sequentially"
proof (unfold f_def, rule Weierstrass_m_test)
fix n :: nat and x :: real assume x: "x ∈ ball 0 1"
{
fix k :: nat assume k: "k ≥ 1"
from x have "x^2 < 1" by (auto simp: dist_0_norm abs_square_less_1)
also from k have "… ≤ of_nat k^2" by simp
finally have "(1 - x^2 / of_nat k^2) ∈ {0..1}" using k
by (simp_all add: field_simps del: of_nat_Suc)
}
hence "(∏k=1..n. abs (1 - x^2 / of_nat k^2)) ≤ (∏k=1..n. 1)" by (intro prod_mono) simp
thus "norm (P x n / (of_nat (Suc n)^2)) ≤ 1 / of_nat (Suc n)^2"
unfolding P_def by (simp add: field_simps abs_prod del: of_nat_Suc)
qed (subst summable_Suc_iff, insert inverse_power_summable[of 2], simp add: inverse_eq_divide)
qed (auto simp: P_def intro!: continuous_intros)
hence "isCont f 0" by (subst (asm) continuous_on_eq_continuous_at) simp_all
hence "(f ─ 0 → f 0)" by (simp add: isCont_def)
also have "f 0 = K" unfolding f_def P_def K_def by (simp add: inverse_eq_divide power_divide)
finally have "f ─ 0 → K" .

moreover have "f ─ 0 → pi^2 / 6"
proof (rule Lim_transform_eventually)
define f' where [abs_def]: "f' x = (∑n. - sin_coeff (n+3) * pi ^ (n+2) * x^n)" for x
have "eventually (λx. x ≠ (0::real)) (at 0)"
by (auto simp add: eventually_at intro!: exI[of _ 1])
thus "eventually (λx. f' x = f x) (at 0)"
proof eventually_elim
fix x :: real assume x: "x ≠ 0"
have "sin_coeff 1 = (1 :: real)" "sin_coeff 2 = (0::real)" by (simp_all add: sin_coeff_def)
with sums_split_initial_segment[OF sums_minus[OF sin_converges], of 3 "pi*x"]
have "(λn. - (sin_coeff (n+3) * (pi*x)^(n+3))) sums (pi * x - sin (pi*x))"
from sums_divide[OF this, of "x^3 * pi"] x
have "(λn. - (sin_coeff (n+3) * pi^(n+2) * x^n)) sums ((1 - sin (pi*x) / (pi*x)) / x^2)"
by (simp add: field_split_simps eval_nat_numeral power_mult_distrib mult_ac)
with x have "(λn. - (sin_coeff (n+3) * pi^(n+2) * x^n)) sums (g x / x^2)"
hence "f' x = g x / x^2" by (simp add: sums_iff f'_def)
also have "… = f x" using sums[of x] x by (simp add: sums_iff g_def f_def)
finally show "f' x = f x" .
qed

have "isCont f' 0" unfolding f'_def
proof (intro isCont_powser_converges_everywhere)
fix x :: real show "summable (λn. -sin_coeff (n+3) * pi^(n+2) * x^n)"
proof (cases "x = 0")
assume x: "x ≠ 0"
from summable_divide[OF sums_summable[OF sums_split_initial_segment[OF
sin_converges[of "pi*x"]], of 3], of "-pi*x^3"] x
show ?thesis by (simp add: mult_ac power_mult_distrib field_split_simps eval_nat_numeral)
qed (simp only: summable_0_powser)
qed
hence "f' ─ 0 → f' 0" by (simp add: isCont_def)
also have "f' 0 = pi * pi / fact 3" unfolding f'_def
by (subst powser_zero) (simp add: sin_coeff_def)
finally show "f' ─ 0 → pi^2 / 6" by (simp add: eval_nat_numeral)
qed

ultimately have "K = pi^2 / 6" by (rule LIM_unique)
moreover from inverse_power_summable[of 2]
have "summable (λn. (inverse (real_of_nat (Suc n)))⇧2)"
by (subst summable_Suc_iff) (simp add: power_inverse)
ultimately```