(* 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 Weierstra{\ss} 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) ultimately show False by contradiction 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" by (simp add: field_simps scaleR_conv_of_real) from sums_mult[OF this, of "inverse z"] and assms show ?thesis by (simp add: field_simps) 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], simp_all add: scaleR_conv_of_real sums_iff sin_coeff_def) 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})" by (simp add: cmod_def) 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})" by (intro real_sqrt_le_mono add_mono) (simp_all add: abs_le_square_iff) 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], simp add: norm_divide inverse_eq_divide) 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)" by (simp add: algebra_simps) 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)" by (simp add: field_simps norm_divide) also have "... ≤ (?z^2 * (of_nat k / (of_nat k - ?z))) / of_nat k^2" using k by (simp add: field_simps power2_eq_square) also have "... ≤ (?z^2 * 2) / of_nat k^2" using k by (intro divide_right_mono mult_left_mono) (simp_all add: field_simps) also note add_divide_distrib [symmetric] 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 by (intro divide_pos_pos mult_pos_pos add_pos_pos) (simp_all add: field_simps) 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 by (simp_all) 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] by (simp add: dist_commute) 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)))" by (simp add: ln_Gamma_series_def algebra_simps) 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 by (simp add: sum_diff) 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})" by (simp add: sum_distrib_left) 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" by (simp add: fact_prod) (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 by (simp add: prod.atLeast0_atMost_Suc atLeastLessThanSuc_atLeastAtMost) 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) 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))" by (simp add: field_split_simps eq_neg_iff_add_eq_0 del: of_nat_Suc) also have "norm … = inverse (of_nat (Suc n)) * inverse (norm (of_nat (Suc n)/t + 1))" by (simp add: norm_divide norm_mult field_split_simps del: of_nat_Suc) 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" by (simp add: Polygamma_def) 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] tendsto_add_filterlim_at_infinity[OF tendsto_const] tendsto_of_nat) 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))" using summable_iff_convergent by blast hence "convergent (λm. (∑n<k. f n) + (∑n<m. f (n + k)))" by (intro convergent_add convergent_const) 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 by (intro le_imp_inverse_le power_mono) (simp_all add: add_ac del: of_nat_Suc) 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] by (simp add: summable_iff_convergent) 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)" by (simp add: Digamma_def summable_sums) 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)) sums Digamma z" by (simp add: add_ac) 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" by (simp add: sums_def sum_subtractf) 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) by (simp add: sums_iff Polygamma_def) 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] tendsto_add_filterlim_at_infinity[OF tendsto_const] tendsto_of_nat) 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) apply (auto simp add: complex_nonpos_Reals_iff) 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))" by (simp add: sums_iff) also from sums summable_deriv_ln_Gamma[OF z''] have "?F' z + … = (∑n. inverse (of_nat (Suc n)) - inverse (z + of_nat n))" by (subst suminf_add) (simp_all add: add_ac sums_iff) 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" by (simp add: Digamma_def) 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]] tendsto_add_filterlim_at_infinity[OF tendsto_const] tendsto_of_nat) 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)))" using summable_Digamma[OF assms] sums by (subst suminf_add) (simp_all add: add_ac sums_iff) 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))" using n by (simp add: Polygamma_def add_ac) 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" by (simp add: harm_Suc) finally show ?case . qed (simp add: harm_def) 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"] by (simp_all add: suminf_of_real) 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" by (simp add: Digamma_def add_ac) 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)))" by (simp_all add: add_ac inverse_mult_distrib[symmetric] ring_distribs del: inverse_divide) 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" using of_nat_neq_0[of "2*n"] by (simp only: of_nat_Suc) (simp add: add_ac) 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 finally show ?case by (simp add: add_ac) 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)" by (simp add: eventually_eventually) 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" by (simp add: rGamma_series_def Gamma_series_def[abs_def]) 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))))" by (subst pochhammer_rec) (simp add: rGamma_series_def field_simps exp_add exp_of_real) also from n have "… = ?f n * rGamma_series z n" by (subst pochhammer_rec') (simp_all add: field_split_simps rGamma_series_def) 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] finally show ?case by (simp add: add_ac pochhammer_rec') 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" by (simp_all add: rGamma_eq_zero_iff Gamma_eq_zero_iff) 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 qed (simp_all add: Gamma_eq_zero_iff nonpos_Ints_of_int) lemma rGamma_of_int: "rGamma (of_int n) = (if n > 0 then inverse (fact (nat (n - 1))) else 0)" by (simp add: Gamma_of_int rGamma_inverse_Gamma) 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) 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) 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))))" by (subst pochhammer_rec) (simp add: rGamma_series_def field_simps exp_add exp_of_real) also from n have "… = ?f n * rGamma_series z n" by (subst pochhammer_rec') (simp_all add: field_split_simps rGamma_series_def add_ac) 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) (simp_all add: convergent_LIMSEQ_iff rGamma_complex_def) 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" by (simp add: rGamma_complex_plus1) 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 has_field_derivative_rGamma_complex_no_nonpos_Int A)+ (simp add: rGamma_complex_1) 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)+ (simp add: algebra_simps rGamma_complex_altdef) 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] 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 Let_def) next fix z :: complex from rGamma_series_complex_converges[of z] have "rGamma_series z ⇢ rGamma z" by (simp add: convergent_LIMSEQ_iff rGamma_complex_def) 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 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] 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 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_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" by (simp add: Gamma_real_pos_exp) lemma Gamma_real_nonneg [simp, intro]: "x > (0::real) ⟹ Gamma x ≥ 0" by (simp add: Gamma_real_pos_exp) 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" by (subst G_fact [symmetric]) (simp add: S_def add_ac of_nat_diff) 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 by (subst (1 2) G_fact [symmetric]) (simp_all add: add_ac of_nat_diff) 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 by (induction n) (simp_all add: pochhammer_Suc add_ac) 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))" by (subst (1 2) G_fact [symmetric]) (simp add: S_def add_ac of_nat_diff) 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 by (simp add: ln_mult) 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 by (induction n) (simp_all add: pochhammer_Suc add_ac) 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"]]) qed (simp_all add: Gamma_series_LIMSEQ) 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"] by (auto simp: add_ac) qed (simp_all add: G_eq_Gamma_aux) 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" by (simp add: frac_def) 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)" using has_field_derivative_Beta1[of y x A] assms by (simp add: Beta_commute add_ac) 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)" by (simp add: Beta_altdef add_divide_distrib algebra_simps) 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" using Beta_plus1_left[of y x] assms by (simp_all add: Beta_commute add.commute) 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"] by (simp add: rGamma_inverse_Gamma) 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)" by (simp add: Gamma_series'_def exp_add ring_distribs power2_eq_square A mult_ac) 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 by (simp add: Gamma_series'_def ln_mult exp_add ring_distribs pochhammer_double) 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)" by (simp add: algebra_simps exp_add[symmetric] add_divide_distrib) 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) (simp_all add: o_def strict_mono_def Gamma_eq_zero_iff) 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) 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"] by (simp add: scaleR_conv_of_real) 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))" by (simp add: field_simps g_def) 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) 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) 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"]) (simp_all add: norm_mult a_def del: of_real_add) 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"]) (simp_all add: norm_mult a_def del: of_real_add) 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) 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 (simp add: h_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) (simp_all add: abs_real_def) 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 (simp add: g_def) 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) 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 (simp add: g_def) 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))" by (simp add: g_def) 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" by (simp add: add_divide_distrib) 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))" by (simp add: add_divide_distrib ac_simps) 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))))" by (simp add: add_ac power2_eq_square exp_add ring_distribs exp_diff exp_of_real) 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] by (simp add: ring_distribs add_divide_distrib ac_simps) 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) 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) 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) 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) 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 by (simp add: pochhammer_same) 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]) (simp_all add: filterlim_at) 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 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) (simp_all add: Gamma_series_euler'_def prod.distrib 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