(* 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'] obtain M where M: "⋀m n. M ≤ m ⟹ norm (∑k = m..n. inverse ((real k)⇧^{2})) < e'" by auto 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] obtain d where d: "0 < d" "⋀t. t ∈ ball z d ⟹ t ∉ ℤ⇩_{≤}⇩_{0}" by auto 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] obtain d where d: "0 < d" "⋀t. t ∈ ball z d ⟹ t ∉ ℤ⇩_{≤}⇩_{0}" by auto 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) lemma higher_deriv_ln_Gamma_complex: assumes "(x::complex) ∉ ℝ⇩_{≤}⇩_{0}" shows "(deriv ^^ j) ln_Gamma x = (if j = 0 then ln_Gamma x else Polygamma (j - 1) x)" proof (cases j) case (Suc j') have "(deriv ^^ j') (deriv ln_Gamma) x = (deriv ^^ j') Digamma x" using eventually_nhds_in_open[of "UNIV - ℝ⇩_{≤}⇩_{0}" x] assms by (intro higher_deriv_cong_ev refl) (auto elim!: eventually_mono simp: open_Diff deriv_ln_Gamma_complex) also have "… = Polygamma j' x" using assms by (subst higher_deriv_Polygamma) (auto elim!: nonpos_Ints_cases simp: complex_nonpos_Reals_iff) finally show ?thesis using Suc by (simp del: funpow.simps add: funpow_Suc_right) qed simp_all 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 </