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exp_local_homeomorph : local_homeomorph ℂ ℂ | local_homeomorph.of_continuous_open
{ to_fun := exp,
inv_fun := log,
source := {z : ℂ | z.im ∈ Ioo (- π) π},
target := {z : ℂ | 0 < z.re} ∪ {z : ℂ | z.im ≠ 0},
map_source' :=
begin
rintro ⟨x, y⟩ ⟨h₁ : -π < y, h₂ : y < π⟩,
refine (not_or_of_imp $ λ hz, _).symm,
obtain rfl : y = 0,
{ r... | def | complex.exp_local_homeomorph | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"exp",
"inv_fun",
"local_homeomorph",
"local_homeomorph.of_continuous_open",
"not_and_distrib",
"not_or_of_imp",
"real.exp_pos",
"real.sin_eq_zero_iff_of_lt_of_lt"
] | `complex.exp` as a `local_homeomorph` with `source = {z | -π < im z < π}` and
`target = {z | 0 < re z} ∪ {z | im z ≠ 0}`. This definition is used to prove that `complex.log`
is complex differentiable at all points but the negative real semi-axis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at_log {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) :
has_strict_deriv_at log x⁻¹ x | have h0 : x ≠ 0, by { rintro rfl, simpa [lt_irrefl] using h },
exp_local_homeomorph.has_strict_deriv_at_symm h h0 $
by simpa [exp_log h0] using has_strict_deriv_at_exp (log x) | lemma | complex.has_strict_deriv_at_log | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_log_real {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) :
has_strict_fderiv_at log (x⁻¹ • (1 : ℂ →L[ℝ] ℂ)) x | (has_strict_deriv_at_log h).complex_to_real_fderiv | lemma | complex.has_strict_fderiv_at_log_real | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_log {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) {n : ℕ∞} :
cont_diff_at ℂ n log x | exp_local_homeomorph.cont_diff_at_symm_deriv (exp_ne_zero $ log x) h
(has_deriv_at_exp _) cont_diff_exp.cont_diff_at | lemma | complex.cont_diff_at_log | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"cont_diff_at",
"has_deriv_at_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E}
(h₁ : has_strict_fderiv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_fderiv_at (λ t, log (f t)) ((f x)⁻¹ • f') x | (has_strict_deriv_at_log h₂).comp_has_strict_fderiv_at x h₁ | lemma | has_strict_fderiv_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : has_strict_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_deriv_at (λ t, log (f t)) (f' / f x) x | by { rw div_eq_inv_mul, exact (has_strict_deriv_at_log h₂).comp x h₁ } | lemma | has_strict_deriv_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.clog_real {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : has_strict_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_strict_deriv_at (λ t, log (f t)) (f' / f x) x | by simpa only [div_eq_inv_mul]
using (has_strict_fderiv_at_log_real h₂).comp_has_strict_deriv_at x h₁ | lemma | has_strict_deriv_at.clog_real | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E}
(h₁ : has_fderiv_at f f' x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_fderiv_at (λ t, log (f t)) ((f x)⁻¹ • f') x | (has_strict_deriv_at_log h₂).has_deriv_at.comp_has_fderiv_at x h₁ | lemma | has_fderiv_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"has_deriv_at.comp_has_fderiv_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : has_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_at (λ t, log (f t)) (f' / f x) x | by { rw div_eq_inv_mul, exact (has_strict_deriv_at_log h₂).has_deriv_at.comp x h₁ } | lemma | has_deriv_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_deriv_at",
"has_deriv_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.clog_real {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : has_deriv_at f f' x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_at (λ t, log (f t)) (f' / f x) x | by simpa only [div_eq_inv_mul]
using (has_strict_fderiv_at_log_real h₂).has_fderiv_at.comp_has_deriv_at x h₁ | lemma | has_deriv_at.clog_real | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_deriv_at",
"has_fderiv_at.comp_has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.clog {f : E → ℂ} {x : E} (h₁ : differentiable_at ℂ f x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_at ℂ (λ t, log (f t)) x | (h₁.has_fderiv_at.clog h₂).differentiable_at | lemma | differentiable_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {s : set E} {x : E}
(h₁ : has_fderiv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_fderiv_within_at (λ t, log (f t)) ((f x)⁻¹ • f') s x | (has_strict_deriv_at_log h₂).has_deriv_at.comp_has_fderiv_within_at x h₁ | lemma | has_fderiv_within_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"has_deriv_at.comp_has_fderiv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.clog {f : ℂ → ℂ} {f' x : ℂ} {s : set ℂ}
(h₁ : has_deriv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_within_at (λ t, log (f t)) (f' / f x) s x | by { rw div_eq_inv_mul,
exact (has_strict_deriv_at_log h₂).has_deriv_at.comp_has_deriv_within_at x h₁ } | lemma | has_deriv_within_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_deriv_at.comp_has_deriv_within_at",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.clog_real {f : ℝ → ℂ} {s : set ℝ} {x : ℝ} {f' : ℂ}
(h₁ : has_deriv_within_at f f' s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
has_deriv_within_at (λ t, log (f t)) (f' / f x) s x | by simpa only [div_eq_inv_mul]
using (has_strict_fderiv_at_log_real h₂).has_fderiv_at.comp_has_deriv_within_at x h₁ | lemma | has_deriv_within_at.clog_real | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_deriv_within_at",
"has_fderiv_at.comp_has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.clog {f : E → ℂ} {s : set E} {x : E}
(h₁ : differentiable_within_at ℂ f s x) (h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_within_at ℂ (λ t, log (f t)) s x | (h₁.has_fderiv_within_at.clog h₂).differentiable_within_at | lemma | differentiable_within_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.clog {f : E → ℂ} {s : set E}
(h₁ : differentiable_on ℂ f s) (h₂ : ∀ x ∈ s, 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable_on ℂ (λ t, log (f t)) s | λ x hx, (h₁ x hx).clog (h₂ x hx) | lemma | differentiable_on.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.clog {f : E → ℂ} (h₁ : differentiable ℂ f)
(h₂ : ∀ x, 0 < (f x).re ∨ (f x).im ≠ 0) :
differentiable ℂ (λ t, log (f t)) | λ x, (h₁ x).clog (h₂ x) | lemma | differentiable.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_integrand_is_o (s : ℝ) :
(λ x:ℝ, exp (-x) * x ^ s) =o[at_top] (λ x:ℝ, exp (-(1/2) * x)) | begin
refine is_o_of_tendsto (λ x hx, _) _,
{ exfalso, exact (exp_pos (-(1 / 2) * x)).ne' hx },
have : (λ (x:ℝ), exp (-x) * x ^ s / exp (-(1 / 2) * x)) = (λ (x:ℝ), exp ((1 / 2) * x) / x ^ s )⁻¹,
{ ext1 x,
field_simp [exp_ne_zero, exp_neg, ← real.exp_add],
left,
ring },
rw this,
exact (tendsto_ex... | lemma | real.Gamma_integrand_is_o | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"exp",
"exp_neg",
"one_half_pos",
"real.exp_add",
"ring",
"tendsto_exp_mul_div_rpow_at_top"
] | Asymptotic bound for the `Γ` function integrand. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_integral_convergent {s : ℝ} (h : 0 < s) :
integrable_on (λ x:ℝ, exp (-x) * x ^ (s - 1)) (Ioi 0) | begin
rw [←Ioc_union_Ioi_eq_Ioi (@zero_le_one ℝ _ _ _ _), integrable_on_union],
split,
{ rw ←integrable_on_Icc_iff_integrable_on_Ioc,
refine integrable_on.continuous_on_mul continuous_on_id.neg.exp _ is_compact_Icc,
refine (interval_integrable_iff_integrable_Icc_of_le zero_le_one).mp _,
exact interval... | lemma | real.Gamma_integral_convergent | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"exp",
"integrable_of_is_O_exp_neg",
"interval_integrable_iff_integrable_Icc_of_le",
"one_half_pos",
"zero_le_one",
"zero_lt_one"
] | The Euler integral for the `Γ` function converges for positive real `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_integral_convergent {s : ℂ} (hs : 0 < s.re) :
integrable_on (λ x, (-x).exp * x ^ (s - 1) : ℝ → ℂ) (Ioi 0) | begin
split,
{ refine continuous_on.ae_strongly_measurable _ measurable_set_Ioi,
apply (continuous_of_real.comp continuous_neg.exp).continuous_on.mul,
apply continuous_at.continuous_on,
intros x hx,
have : continuous_at (λ x:ℂ, x ^ (s - 1)) ↑x,
{ apply continuous_at_cpow_const, rw of_real_re, ex... | lemma | complex.Gamma_integral_convergent | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"abs_of_nonneg",
"continuous_at",
"continuous_at.comp",
"continuous_at.continuous_on",
"continuous_at_cpow_const",
"continuous_on.ae_strongly_measurable",
"continuous_on.mul",
"exp",
"map_mul",
"measurable_set_Ioi",
"real.Gamma_integral_convergent"
] | The integral defining the `Γ` function converges for complex `s` with `0 < re s`.
This is proved by reduction to the real case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_integral (s : ℂ) : ℂ | ∫ x in Ioi (0:ℝ), ↑(-x).exp * ↑x ^ (s - 1) | def | complex.Gamma_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"exp"
] | Euler's integral for the `Γ` function (of a complex variable `s`), defined as
`∫ x in Ioi 0, exp (-x) * x ^ (s - 1)`.
See `complex.Gamma_integral_convergent` for a proof of the convergence of the integral for
`0 < re s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_integral_conj (s : ℂ) : Gamma_integral (conj s) = conj (Gamma_integral s) | begin
rw [Gamma_integral, Gamma_integral, ←integral_conj],
refine set_integral_congr measurable_set_Ioi (λ x hx, _),
dsimp only,
rw [ring_hom.map_mul, conj_of_real, cpow_def_of_ne_zero (of_real_ne_zero.mpr (ne_of_gt hx)),
cpow_def_of_ne_zero (of_real_ne_zero.mpr (ne_of_gt hx)), ←exp_conj, ring_hom.map_mul,
... | lemma | complex.Gamma_integral_conj | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"measurable_set_Ioi",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_integral_of_real (s : ℝ) :
Gamma_integral ↑s = ↑(∫ x:ℝ in Ioi 0, real.exp (-x) * x ^ (s - 1)) | begin
rw [Gamma_integral, ←_root_.integral_of_real],
refine set_integral_congr measurable_set_Ioi _,
intros x hx, dsimp only,
rw [of_real_mul, of_real_cpow (mem_Ioi.mp hx).le],
simp,
end | lemma | complex.Gamma_integral_of_real | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"measurable_set_Ioi",
"real.exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_integral_one : Gamma_integral 1 = 1 | by simpa only [←of_real_one, Gamma_integral_of_real, of_real_inj, sub_self,
rpow_zero, mul_one] using integral_exp_neg_Ioi_zero | lemma | complex.Gamma_integral_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"integral_exp_neg_Ioi_zero",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_Gamma (s : ℂ) (X : ℝ) : ℂ | ∫ x in 0..X, (-x).exp * x ^ (s - 1) | def | complex.partial_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"exp"
] | The indefinite version of the `Γ` function, `Γ(s, X) = ∫ x ∈ 0..X, exp(-x) x ^ (s - 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_partial_Gamma {s : ℂ} (hs: 0 < s.re) :
tendsto (λ X:ℝ, partial_Gamma s X) at_top (𝓝 $ Gamma_integral s) | interval_integral_tendsto_integral_Ioi 0 (Gamma_integral_convergent hs) tendsto_id | lemma | complex.tendsto_partial_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_integrand_interval_integrable (s : ℂ) {X : ℝ} (hs : 0 < s.re) (hX : 0 ≤ X):
interval_integrable (λ x, (-x).exp * x ^ (s - 1) : ℝ → ℂ) volume 0 X | begin
rw interval_integrable_iff_integrable_Ioc_of_le hX,
exact integrable_on.mono_set (Gamma_integral_convergent hs) Ioc_subset_Ioi_self
end | lemma | complex.Gamma_integrand_interval_integrable | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"exp",
"interval_integrable",
"interval_integrable_iff_integrable_Ioc_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_integrand_deriv_integrable_A {s : ℂ} (hs : 0 < s.re) {X : ℝ} (hX : 0 ≤ X):
interval_integrable (λ x, -((-x).exp * x ^ s) : ℝ → ℂ) volume 0 X | begin
convert (Gamma_integrand_interval_integrable (s+1) _ hX).neg,
{ ext1, simp only [add_sub_cancel, pi.neg_apply] },
{ simp only [add_re, one_re], linarith,},
end | lemma | complex.Gamma_integrand_deriv_integrable_A | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"exp",
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_integrand_deriv_integrable_B {s : ℂ} (hs : 0 < s.re) {Y : ℝ} (hY : 0 ≤ Y) :
interval_integrable (λ (x : ℝ), (-x).exp * (s * x ^ (s - 1)) : ℝ → ℂ) volume 0 Y | begin
have : (λ x, (-x).exp * (s * x ^ (s - 1)) : ℝ → ℂ) =
(λ x, s * ((-x).exp * x ^ (s - 1)) : ℝ → ℂ),
{ ext1, ring, },
rw [this, interval_integrable_iff_integrable_Ioc_of_le hY],
split,
{ refine (continuous_on_const.mul _).ae_strongly_measurable measurable_set_Ioc,
apply (continuous_of_real.comp con... | lemma | complex.Gamma_integrand_deriv_integrable_B | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"abs_of_nonneg",
"continuous_at",
"continuous_at.continuous_on",
"continuous_at_cpow_const",
"continuous_on.mul",
"exp",
"interval_integrable",
"interval_integrable_iff_integrable_Ioc_of_le",
"map_mul",
"measurable_set_Ioc",
"real.Gamma_integral_convergent",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_Gamma_add_one {s : ℂ} (hs: 0 < s.re) {X : ℝ} (hX : 0 ≤ X) :
partial_Gamma (s + 1) X = s * partial_Gamma s X - (-X).exp * X ^ s | begin
rw [partial_Gamma, partial_Gamma, add_sub_cancel],
have F_der_I: (∀ (x:ℝ), (x ∈ Ioo 0 X) → has_deriv_at (λ x, (-x).exp * x ^ s : ℝ → ℂ)
( -((-x).exp * x ^ s) + (-x).exp * (s * x ^ (s - 1))) x),
{ intros x hx,
have d1 : has_deriv_at (λ (y: ℝ), (-y).exp) (-(-x).exp) x,
{ simpa using (has_deriv_at_... | lemma | complex.partial_Gamma_add_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"cont",
"exp",
"has_deriv_at",
"has_deriv_at.cpow_const",
"has_deriv_at_id",
"has_deriv_at_neg",
"interval_integral.integral_add",
"interval_integral.integral_neg",
"mul_one",
"mul_zero",
"neg_mul",
"ring"
] | The recurrence relation for the indefinite version of the `Γ` function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_integral_add_one {s : ℂ} (hs: 0 < s.re) :
Gamma_integral (s + 1) = s * Gamma_integral s | begin
suffices : tendsto (s+1).partial_Gamma at_top (𝓝 $ s * Gamma_integral s),
{ refine tendsto_nhds_unique _ this,
apply tendsto_partial_Gamma, rw [add_re, one_re], linarith, },
have : (λ X:ℝ, s * partial_Gamma s X - X ^ s * (-X).exp) =ᶠ[at_top] (s+1).partial_Gamma,
{ apply eventually_eq_of_mem (Ici_mem_... | theorem | complex.Gamma_integral_add_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"abs_of_nonneg",
"exp",
"map_mul",
"neg_mul",
"one_mul",
"tendsto_nhds_unique",
"tendsto_rpow_mul_exp_neg_mul_at_top_nhds_0",
"zero_lt_one"
] | The recurrence relation for the `Γ` integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_aux : ℕ → (ℂ → ℂ) | | 0 := Gamma_integral
| (n+1) := λ s:ℂ, (Gamma_aux n (s+1)) / s | def | complex.Gamma_aux | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [] | The `n`th function in this family is `Γ(s)` if `-n < s.re`, and junk otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_aux_recurrence1 (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) :
Gamma_aux n s = Gamma_aux n (s+1) / s | begin
induction n with n hn generalizing s,
{ simp only [nat.cast_zero, neg_lt_zero] at h1,
dsimp only [Gamma_aux], rw Gamma_integral_add_one h1,
rw [mul_comm, mul_div_cancel], contrapose! h1, rw h1,
simp },
{ dsimp only [Gamma_aux],
have hh1 : -(s+1).re < n,
{ rw [nat.succ_eq_add_one, nat.cas... | lemma | complex.Gamma_aux_recurrence1 | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"mul_comm",
"mul_div_cancel",
"nat.cast_add",
"nat.cast_one",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_aux_recurrence2 (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) :
Gamma_aux n s = Gamma_aux (n+1) s | begin
cases n,
{ simp only [nat.cast_zero, neg_lt_zero] at h1,
dsimp only [Gamma_aux],
rw [Gamma_integral_add_one h1, mul_div_cancel_left],
rintro rfl,
rw [zero_re] at h1,
exact h1.false },
{ dsimp only [Gamma_aux],
have : (Gamma_aux n (s + 1 + 1)) / (s+1) = Gamma_aux n (s + 1),
{ have... | lemma | complex.Gamma_aux_recurrence2 | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"mul_div_cancel_left",
"nat.cast_add",
"nat.cast_one",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma (s : ℂ) : ℂ | Gamma_aux ⌊1 - s.re⌋₊ s | def | complex.Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma"
] | The `Γ` function (of a complex variable `s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_eq_Gamma_aux (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) : Gamma s = Gamma_aux n s | begin
have u : ∀ (k : ℕ), Gamma_aux (⌊1 - s.re⌋₊ + k) s = Gamma s,
{ intro k, induction k with k hk,
{ simp [Gamma],},
{ rw [←hk, nat.succ_eq_add_one, ←add_assoc],
refine (Gamma_aux_recurrence2 s (⌊1 - s.re⌋₊ + k) _).symm,
rw nat.cast_add,
have i0 := nat.sub_one_lt_floor (1 - s.re),
... | lemma | complex.Gamma_eq_Gamma_aux | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"nat.cast_add",
"nat.cast_le",
"nat.floor_le",
"nat.floor_of_nonpos",
"nat.sub_one_lt_floor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_add_one (s : ℂ) (h2 : s ≠ 0) : Gamma (s+1) = s * Gamma s | begin
let n := ⌊1 - s.re⌋₊,
have t1 : -s.re < n,
{ simpa only [sub_sub_cancel_left] using nat.sub_one_lt_floor (1 - s.re) },
have t2 : -(s+1).re < n,
{ rw [add_re, one_re], linarith, },
rw [Gamma_eq_Gamma_aux s n t1, Gamma_eq_Gamma_aux (s+1) n t2, Gamma_aux_recurrence1 s n t1],
field_simp, ring,
end | theorem | complex.Gamma_add_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"nat.sub_one_lt_floor",
"ring"
] | The recurrence relation for the `Γ` function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_eq_integral {s : ℂ} (hs : 0 < s.re) : Gamma s = Gamma_integral s | Gamma_eq_Gamma_aux s 0 (by { norm_cast, linarith }) | theorem | complex.Gamma_eq_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_one : Gamma 1 = 1 | by { rw Gamma_eq_integral, simpa using Gamma_integral_one, simp } | lemma | complex.Gamma_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_nat_eq_factorial (n : ℕ) : Gamma (n+1) = n! | begin
induction n with n hn,
{ simpa using Gamma_one },
{ rw (Gamma_add_one n.succ $ nat.cast_ne_zero.mpr $ nat.succ_ne_zero n),
simp only [nat.cast_succ, nat.factorial_succ, nat.cast_mul], congr, exact hn },
end | theorem | complex.Gamma_nat_eq_factorial | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"nat.cast_mul",
"nat.cast_succ",
"nat.factorial_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_zero : Gamma 0 = 0 | by simp_rw [Gamma, zero_re, sub_zero, nat.floor_one, Gamma_aux, div_zero] | lemma | complex.Gamma_zero | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"div_zero",
"nat.floor_one"
] | At `0` the Gamma function is undefined; by convention we assign it the value `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_neg_nat_eq_zero (n : ℕ) : Gamma (-n) = 0 | begin
induction n with n IH,
{ rw [nat.cast_zero, neg_zero, Gamma_zero] },
{ have A : -(n.succ : ℂ) ≠ 0,
{ rw [neg_ne_zero, nat.cast_ne_zero],
apply nat.succ_ne_zero },
have : -(n:ℂ) = -↑n.succ + 1, by simp,
rw [this, Gamma_add_one _ A] at IH,
contrapose! IH,
exact mul_ne_zero A IH }
end | lemma | complex.Gamma_neg_nat_eq_zero | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"mul_ne_zero",
"nat.cast_ne_zero",
"nat.cast_zero"
] | At `-n` for `n ∈ ℕ`, the Gamma function is undefined; by convention we assign it the value 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_conj (s : ℂ) : Gamma (conj s) = conj (Gamma s) | begin
suffices : ∀ (n:ℕ) (s:ℂ) , Gamma_aux n (conj s) = conj (Gamma_aux n s), from this _ _,
intro n,
induction n with n IH,
{ rw Gamma_aux, exact Gamma_integral_conj, },
{ intro s,
rw Gamma_aux,
dsimp only,
rw [div_eq_mul_inv _ s, ring_hom.map_mul, conj_inv, ←div_eq_mul_inv],
suffices : conj ... | lemma | complex.Gamma_conj | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"conj_inv",
"div_eq_mul_inv",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_integral_eq_mellin : Gamma_integral = mellin (λ x, real.exp (-x)) | funext (λ s, by simp only [mellin, Gamma_integral, smul_eq_mul, mul_comm]) | lemma | complex.Gamma_integral_eq_mellin | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"mellin",
"mul_comm",
"real.exp",
"smul_eq_mul"
] | Rewrite the Gamma integral as an example of a Mellin transform. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_Gamma_integral {s : ℂ} (hs : 0 < s.re) :
has_deriv_at Gamma_integral (∫ (t : ℝ) in Ioi 0, t ^ (s - 1) * (real.log t * real.exp (-t))) s | begin
rw Gamma_integral_eq_mellin,
convert (mellin_has_deriv_of_is_O_rpow _ _ (lt_add_one _) _ hs).2,
{ refine (continuous.continuous_on _).locally_integrable_on measurable_set_Ioi,
exact continuous_of_real.comp (real.continuous_exp.comp continuous_neg), },
{ rw [←is_O_norm_left],
simp_rw [complex.norm_... | theorem | complex.has_deriv_at_Gamma_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"complex.norm_eq_abs",
"continuous.continuous_on",
"continuous_within_at",
"has_deriv_at",
"is_o_exp_neg_mul_rpow_at_top",
"lt_add_one",
"measurable_set_Ioi",
"mellin_has_deriv_of_is_O_rpow",
"neg_one_mul",
"one_ne_zero",
"real.exp",
"real.log",
"zero_lt_one"
] | The derivative of the `Γ` integral, at any `s ∈ ℂ` with `1 < re s`, is given by the Melllin
transform of `log t * exp (-t)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at_Gamma_aux (s : ℂ) (n : ℕ) (h1 : (1 - s.re) < n ) (h2 : ∀ m : ℕ, s ≠ -m) :
differentiable_at ℂ (Gamma_aux n) s | begin
induction n with n hn generalizing s,
{ refine (has_deriv_at_Gamma_integral _).differentiable_at,
rw nat.cast_zero at h1, linarith },
{ dsimp only [Gamma_aux],
specialize hn (s + 1),
have a : 1 - (s + 1).re < ↑n,
{ rw nat.cast_succ at h1, rw [complex.add_re, complex.one_re], linarith },
... | lemma | complex.differentiable_at_Gamma_aux | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"complex.add_re",
"complex.one_re",
"differentiable_at",
"differentiable_at.comp",
"differentiable_at.div",
"nat.cast_succ",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_Gamma (s : ℂ) (hs : ∀ m : ℕ, s ≠ -m) : differentiable_at ℂ Gamma s | begin
let n := ⌊1 - s.re⌋₊ + 1,
have hn : 1 - s.re < n := by exact_mod_cast nat.lt_floor_add_one (1 - s.re),
apply (differentiable_at_Gamma_aux s n hn hs).congr_of_eventually_eq,
let S := { t : ℂ | 1 - t.re < n },
have : S ∈ 𝓝 s,
{ rw mem_nhds_iff, use S,
refine ⟨subset.rfl, _, hn⟩,
have : S = re⁻¹... | theorem | complex.differentiable_at_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"differentiable_at",
"is_open_Ioi",
"mem_nhds_iff",
"nat.lt_floor_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_self_mul_Gamma_nhds_zero : tendsto (λ z : ℂ, z * Gamma z) (𝓝[≠] 0) (𝓝 1) | begin
rw (show 𝓝 (1 : ℂ) = 𝓝 (Gamma (0 + 1)), by simp only [zero_add, complex.Gamma_one]),
convert (tendsto.mono_left _ nhds_within_le_nhds).congr'
(eventually_eq_of_mem self_mem_nhds_within complex.Gamma_add_one),
refine continuous_at.comp _ (continuous_id.add continuous_const).continuous_at,
refine (com... | lemma | complex.tendsto_self_mul_Gamma_nhds_zero | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma_add_one",
"complex.Gamma_one",
"complex.differentiable_at_Gamma",
"continuous_at",
"continuous_at.comp",
"continuous_const",
"nat.cast_nonneg",
"nhds_within_le_nhds",
"self_mem_nhds_within",
"zero_lt_one"
] | At `s = 0`, the Gamma function has a simple pole with residue 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma (s : ℝ) : ℝ | (complex.Gamma s).re | def | real.Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma"
] | The `Γ` function (of a real variable `s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_eq_integral {s : ℝ} (hs : 0 < s) : Gamma s = ∫ x in Ioi 0, exp (-x) * x ^ (s - 1) | begin
rw [Gamma, complex.Gamma_eq_integral (by rwa complex.of_real_re : 0 < complex.re s)],
dsimp only [complex.Gamma_integral],
simp_rw [←complex.of_real_one, ←complex.of_real_sub],
suffices : ∫ (x : ℝ) in Ioi 0, ↑(exp (-x)) * (x : ℂ) ^ ((s - 1 : ℝ) : ℂ) =
∫ (x : ℝ) in Ioi 0, ((exp (-x) * x ^ (s - 1) : ℝ) ... | lemma | real.Gamma_eq_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma_eq_integral",
"complex.Gamma_integral",
"complex.of_real_cpow",
"complex.of_real_re",
"exp",
"measurable_set_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_add_one {s : ℝ} (hs : s ≠ 0) : Gamma (s + 1) = s * Gamma s | begin
simp_rw Gamma,
rw [complex.of_real_add, complex.of_real_one, complex.Gamma_add_one, complex.of_real_mul_re],
rwa complex.of_real_ne_zero,
end | lemma | real.Gamma_add_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma_add_one",
"complex.of_real_add",
"complex.of_real_mul_re",
"complex.of_real_ne_zero",
"complex.of_real_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_one : Gamma 1 = 1 | by rw [Gamma, complex.of_real_one, complex.Gamma_one, complex.one_re] | lemma | real.Gamma_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma_one",
"complex.of_real_one",
"complex.one_re"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.complex.Gamma_of_real (s : ℝ) : complex.Gamma (s : ℂ) = Gamma s | by rw [Gamma, eq_comm, ←complex.conj_eq_iff_re, ←complex.Gamma_conj, complex.conj_of_real] | lemma | complex.Gamma_of_real | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma",
"complex.conj_of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_nat_eq_factorial (n : ℕ) : Gamma (n + 1) = n! | by rw [Gamma, complex.of_real_add, complex.of_real_nat_cast, complex.of_real_one,
complex.Gamma_nat_eq_factorial, ←complex.of_real_nat_cast, complex.of_real_re] | theorem | real.Gamma_nat_eq_factorial | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma_nat_eq_factorial",
"complex.of_real_add",
"complex.of_real_nat_cast",
"complex.of_real_one",
"complex.of_real_re"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_zero : Gamma 0 = 0 | by simpa only [←complex.of_real_zero, complex.Gamma_of_real, complex.of_real_inj]
using complex.Gamma_zero | lemma | real.Gamma_zero | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma_of_real",
"complex.Gamma_zero",
"complex.of_real_inj"
] | At `0` the Gamma function is undefined; by convention we assign it the value `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_neg_nat_eq_zero (n : ℕ) : Gamma (-n) = 0 | begin
simpa only [←complex.of_real_nat_cast, ←complex.of_real_neg, complex.Gamma_of_real,
complex.of_real_eq_zero] using complex.Gamma_neg_nat_eq_zero n,
end | lemma | real.Gamma_neg_nat_eq_zero | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.Gamma_neg_nat_eq_zero",
"complex.Gamma_of_real",
"complex.of_real_eq_zero"
] | At `-n` for `n ∈ ℕ`, the Gamma function is undefined; by convention we assign it the value `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_pos_of_pos {s : ℝ} (hs : 0 < s) : 0 < Gamma s | begin
rw Gamma_eq_integral hs,
have : function.support (λ (x : ℝ), exp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0,
{ rw inter_eq_right_iff_subset,
intros x hx,
rw function.mem_support,
exact mul_ne_zero (exp_pos _).ne' (rpow_pos_of_pos hx _).ne' },
rw set_integral_pos_iff_support_of_nonneg_ae,
{ rw [this... | lemma | real.Gamma_pos_of_pos | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"ennreal.of_real_lt_top",
"exp",
"function.support",
"measurable_set_Ioi",
"mul_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_ne_zero {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0 | begin
suffices : ∀ {n : ℕ}, (-(n:ℝ) < s) → Gamma s ≠ 0,
{ apply this,
swap, use (⌊-s⌋₊ + 1),
rw [neg_lt, nat.cast_add, nat.cast_one],
exact nat.lt_floor_add_one _ },
intro n,
induction n generalizing s,
{ intro hs,
refine (Gamma_pos_of_pos _).ne',
rwa [nat.cast_zero, neg_zero] at hs },
{... | lemma | real.Gamma_ne_zero | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"mul_ne_zero_iff",
"nat.cast_add",
"nat.cast_one",
"nat.cast_zero",
"nat.lt_floor_add_one",
"ring"
] | The Gamma function does not vanish on `ℝ` (except at non-positive integers, where the function
is mathematically undefined and we set it to `0` by convention). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_eq_zero_iff (s : ℝ) : Gamma s = 0 ↔ ∃ m : ℕ, s = -m | ⟨by { contrapose!, exact Gamma_ne_zero }, by { rintro ⟨m, rfl⟩, exact Gamma_neg_nat_eq_zero m }⟩ | lemma | real.Gamma_eq_zero_iff | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_Gamma {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : differentiable_at ℝ Gamma s | begin
refine ((complex.differentiable_at_Gamma _ _).has_deriv_at).real_of_complex.differentiable_at,
simp_rw [←complex.of_real_nat_cast, ←complex.of_real_neg, ne.def, complex.of_real_inj],
exact hs,
end | lemma | real.differentiable_at_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/basic.lean | [
"measure_theory.integral.exp_decay",
"analysis.special_functions.improper_integrals",
"analysis.mellin_transform"
] | [
"Gamma",
"complex.differentiable_at_Gamma",
"complex.of_real_inj",
"differentiable_at",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
beta_integral (u v : ℂ) : ℂ | ∫ (x:ℝ) in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1) | def | complex.beta_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [] | The Beta function `Β (u, v)`, defined as `∫ x:ℝ in 0..1, x ^ (u - 1) * (1 - x) ^ (v - 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
beta_integral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
interval_integrable (λ x, x ^ (u - 1) * (1 - x) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | begin
apply interval_integrable.mul_continuous_on,
{ refine interval_integral.interval_integrable_cpow' _,
rwa [sub_re, one_re, ←zero_sub, sub_lt_sub_iff_right] },
{ apply continuous_at.continuous_on,
intros x hx,
rw uIcc_of_le (by positivity: (0:ℝ) ≤ 1/2) at hx,
apply continuous_at.cpow,
{ ex... | lemma | complex.beta_integral_convergent_left | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"continuous_at",
"continuous_at.continuous_on",
"continuous_at.cpow",
"continuous_at_const",
"interval_integrable",
"interval_integrable.mul_continuous_on",
"interval_integral.interval_integrable_cpow'"
] | Auxiliary lemma for `beta_integral_convergent`, showing convergence at the left endpoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
beta_integral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
interval_integrable (λ x, x ^ (u - 1) * (1 - x) ^ (v - 1) : ℝ → ℂ) volume 0 1 | begin
refine (beta_integral_convergent_left hu v).trans _,
rw interval_integrable.iff_comp_neg,
convert ((beta_integral_convergent_left hv u).comp_add_right 1).symm,
{ ext1 x,
conv_lhs { rw mul_comm },
congr' 2;
{ push_cast, ring } },
{ norm_num },
{ norm_num }
end | lemma | complex.beta_integral_convergent | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"interval_integrable",
"interval_integrable.iff_comp_neg",
"mul_comm",
"ring"
] | The Beta integral is convergent for all `u, v` of positive real part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
beta_integral_symm (u v : ℂ) :
beta_integral v u = beta_integral u v | begin
rw [beta_integral, beta_integral],
have := interval_integral.integral_comp_mul_add
(λ x:ℝ, (x:ℂ) ^ (u - 1) * (1 - ↑x) ^ (v - 1)) (neg_one_lt_zero.ne) 1,
rw [inv_neg, inv_one, neg_one_smul, ←interval_integral.integral_symm] at this,
convert this,
{ ext1 x, rw mul_comm, congr;
{ push_cast, ring } ... | lemma | complex.beta_integral_symm | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"interval_integral.integral_comp_mul_add",
"inv_neg",
"inv_one",
"mul_comm",
"neg_one_smul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
beta_integral_eval_one_right {u : ℂ} (hu : 0 < re u) :
beta_integral u 1 = 1 / u | begin
simp_rw [beta_integral, sub_self, cpow_zero, mul_one],
rw integral_cpow (or.inl _),
{ rw [of_real_zero, of_real_one, one_cpow, zero_cpow,
sub_zero, sub_add_cancel],
rw sub_add_cancel,
contrapose! hu, rw [hu, zero_re] },
{ rwa [sub_re, one_re, ←sub_pos, sub_neg_eq_add, sub_add_cancel] },
end | lemma | complex.beta_integral_eval_one_right | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"integral_cpow",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
beta_integral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in 0..a, (x:ℂ) ^ (s - 1) * (a - x) ^ (t - 1) = a ^ (s + t - 1) * beta_integral s t | begin
have ha' : (a:ℂ) ≠ 0, from of_real_ne_zero.mpr ha.ne',
rw beta_integral,
have A : (a:ℂ) ^ (s + t - 1) = a * (a ^ (s - 1) * a ^ (t - 1)),
{ rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)),
cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one, mul_assoc] },
rw [A, mul_assoc, ←interval_integral.integral_co... | lemma | complex.beta_integral_scaled | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"div_le_one",
"div_pos",
"interval_integral.integral_of_le",
"measurable_set_Ioc",
"mul_assoc",
"mul_div_cancel'",
"mul_mul_mul_comm",
"mul_one",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_mul_Gamma_eq_beta_integral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * beta_integral s t | begin
-- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
-- this as a formula for the Beta function.
have conv_int := integral_pos_convolution (Gamma_integral_convergent hs)
(Gamma_integral_convergent ht) (continuous_linear_map.mul ℝ ℂ),
simp_rw continuous_line... | lemma | complex.Gamma_mul_Gamma_eq_beta_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"continuous_linear_map.mul",
"continuous_linear_map.mul_apply'",
"integral_pos_convolution",
"measurable_set_Ioi",
"mul_assoc",
"ring"
] | Relation between Beta integral and Gamma function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
beta_integral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
u * beta_integral u (v + 1) = v * beta_integral (u + 1) v | begin
-- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of
-- `Gamma_mul_Gamma_eq_beta_integral`; but we don't know that yet. We will prove it later, but
-- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument.
let F : ℝ → ℂ := λ x, x ^ u * (1 - x) ^ v,
... | lemma | complex.beta_integral_recurrence | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"continuous_at",
"continuous_at.continuous_on",
"continuous_on",
"has_deriv_at",
"has_deriv_at.comp",
"has_deriv_at.const_sub",
"has_deriv_at.cpow_const",
"has_deriv_at_id",
"interval_integrable.const_mul",
"interval_integral.integral_const_mul",
"interval_integral.integral_eq_sub_of_has_deriv_a... | Recurrence formula for the Beta function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
beta_integral_eval_nat_add_one_right {u : ℂ} (hu : 0 < re u) (n : ℕ) :
beta_integral u (n + 1) = n! / ∏ (j:ℕ) in finset.range (n + 1), (u + j) | begin
induction n with n IH generalizing u,
{ rw [nat.cast_zero, zero_add, beta_integral_eval_one_right hu,
nat.factorial_zero, nat.cast_one, zero_add, finset.prod_range_one, nat.cast_zero, add_zero] },
{ have := beta_integral_recurrence hu (_ : 0 < re n.succ),
swap, { rw [←of_real_nat_cast, of_real_re]... | lemma | complex.beta_integral_eval_nat_add_one_right | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"finset.prod_range_one",
"finset.prod_range_succ'",
"finset.range",
"mul_comm",
"nat.cast_add",
"nat.cast_mul",
"nat.cast_one",
"nat.cast_succ",
"nat.cast_zero",
"nat.factorial_succ",
"nat.factorial_zero"
] | Explicit formula for the Beta function when second argument is a positive integer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_seq (s : ℂ) (n : ℕ) | (n:ℂ) ^ s * n! / ∏ (j:ℕ) in finset.range (n + 1), (s + j) | def | complex.Gamma_seq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"finset.range"
] | The sequence with `n`-th term `n ^ s * n! / (s * (s + 1) * ... * (s + n))`, for complex `s`.
We will show that this tends to `Γ(s)` as `n → ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_seq_eq_beta_integral_of_re_pos {s : ℂ} (hs : 0 < re s) (n : ℕ) :
Gamma_seq s n = n ^ s * beta_integral s (n + 1) | by rw [Gamma_seq, beta_integral_eval_nat_add_one_right hs n, ←mul_div_assoc] | lemma | complex.Gamma_seq_eq_beta_integral_of_re_pos | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_seq_add_one_left (s : ℂ) {n : ℕ} (hn : n ≠ 0) :
(Gamma_seq (s + 1) n) / s = n / (n + 1 + s) * Gamma_seq s n | begin
conv_lhs { rw [Gamma_seq, finset.prod_range_succ, div_div] },
conv_rhs { rw [Gamma_seq, finset.prod_range_succ', nat.cast_zero, add_zero, div_mul_div_comm,
←mul_assoc, ←mul_assoc, mul_comm _ (finset.prod _ _)] },
congr' 3,
{ rw [cpow_add _ _ (nat.cast_ne_zero.mpr hn), cpow_one, mul_comm] },
{ refine... | lemma | complex.Gamma_seq_add_one_left | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"div_div",
"div_mul_div_comm",
"finset.prod",
"finset.prod_congr",
"finset.prod_range_succ",
"finset.prod_range_succ'",
"mul_comm",
"nat.cast_zero",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_seq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (hn : n ≠ 0) :
Gamma_seq s n = ∫ x:ℝ in 0..n, ↑((1 - x / n) ^ n) * (x:ℂ) ^ (s - 1) | begin
have : ∀ (x : ℝ), x = x / n * n, by { intro x, rw div_mul_cancel, exact nat.cast_ne_zero.mpr hn },
conv in (↑_ ^ _) { congr, rw this x },
rw Gamma_seq_eq_beta_integral_of_re_pos hs,
rw [beta_integral, @interval_integral.integral_comp_div _ _ _ _ 0 n _
(λ x, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1) : ℝ → ℂ... | lemma | complex.Gamma_seq_eq_approx_Gamma_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"div_mul_cancel",
"div_self",
"interval_integral.integral_comp_div",
"interval_integral.integral_of_le",
"measurable_set_Ioc",
"ring",
"zero_div",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
tendsto (λ n:ℕ, ∫ x:ℝ in 0..n, ↑((1 - x / n) ^ n) * (x:ℂ) ^ (s - 1)) at_top (𝓝 $ Gamma s) | begin
rw [Gamma_eq_integral hs],
-- We apply dominated convergence to the following function, which we will show is uniformly
-- bounded above by the Gamma integrand `exp (-x) * x ^ (re s - 1)`.
let f : ℕ → ℝ → ℂ := λ n, indicator (Ioc 0 (n:ℝ))
(λ x:ℝ, ↑((1 - x / n) ^ n) * (x:ℂ) ^ (s - 1)),
-- integrabili... | lemma | complex.approx_Gamma_integral_tendsto_Gamma_integral | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"complex.abs_of_nonneg",
"complex.norm_eq_abs",
"continuity",
"continuous.continuous_on",
"div_le_one_of_le",
"interval_integrable.continuous_on_mul",
"interval_integral.integral_of_le",
"interval_integral.interval_integrable_cpow'",
"measurable_set",
"measurable_set_Ioc",
"measurable... | The main techical lemma for `Gamma_seq_tendsto_Gamma`, expressing the integral defining the
Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_seq_tendsto_Gamma (s : ℂ) :
tendsto (Gamma_seq s) at_top (𝓝 $ Gamma s) | begin
suffices : ∀ m : ℕ, (-↑m < re s) → tendsto (Gamma_seq s) at_top (𝓝 $ Gamma_aux m s),
{ rw Gamma,
apply this,
rw neg_lt,
rcases lt_or_le 0 (re s) with hs | hs,
{ exact (neg_neg_of_pos hs).trans_le (nat.cast_nonneg _), },
{ refine (nat.lt_floor_add_one _).trans_le _,
rw [sub_eq_neg_ad... | lemma | complex.Gamma_seq_tendsto_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"mul_comm",
"nat.cast_add_one",
"nat.cast_nonneg",
"nat.cast_succ",
"nat.cast_zero",
"nat.floor_add_one",
"nat.lt_floor_add_one",
"one_ne_zero'",
"tendsto_coe_nat_div_add_at_top"
] | Euler's limit formula for the complex Gamma function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_seq_mul (z : ℂ) {n : ℕ} (hn : n ≠ 0) :
Gamma_seq z n * Gamma_seq (1 - z) n =
n / (n + 1 - z) * (1 / (z * ∏ j in finset.range n, (1 - z ^ 2 / (j + 1) ^ 2))) | begin
-- also true for n = 0 but we don't need it
have aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d), by { intros, ring },
rw [Gamma_seq, Gamma_seq, div_mul_div_comm, aux, ←pow_two],
have : (n : ℂ) ^ z * n ^ (1 - z) = n,
{ rw [←cpow_add _ _ (nat.cast_ne_zero.mpr hn), add_sub_cancel'_right, cpow_one... | lemma | complex.Gamma_seq_mul | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"aux",
"div_mul_div_comm",
"finset.prod_mul_distrib",
"finset.prod_pow",
"finset.prod_range_add_one_eq_factorial",
"finset.prod_range_succ",
"finset.prod_range_succ'",
"finset.range",
"mul_comm",
"mul_div_cancel",
"mul_one_div",
"nat.cast_ne_zero",
"nat.cast_pow",
"nat.cast_zero",
"nat.f... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_mul_Gamma_one_sub (z : ℂ) : Gamma z * Gamma (1 - z) = π / sin (π * z) | begin
have pi_ne : (π : ℂ) ≠ 0, from complex.of_real_ne_zero.mpr pi_ne_zero,
by_cases hs : sin (↑π * z) = 0,
{ -- first deal with silly case z = integer
rw [hs, div_zero],
rw [←neg_eq_zero, ←complex.sin_neg, ←mul_neg, complex.sin_eq_zero_iff, mul_comm] at hs,
obtain ⟨k, hk⟩ := hs,
rw [mul_eq_mul_r... | theorem | complex.Gamma_mul_Gamma_one_sub | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"complex.Gamma_neg_nat_eq_zero",
"complex.sin_eq_zero_iff",
"div_mul_cancel",
"div_ne_zero",
"div_zero",
"int.cast_neg_succ_of_nat",
"int.cast_of_nat",
"mul_comm",
"mul_eq_mul_right_iff",
"mul_zero",
"nat.cast_add",
"nat.cast_one",
"one_mul",
"tendsto_coe_nat_div_add_at_top",
... | Euler's reflection formula for the complex Gamma function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_ne_zero {s : ℂ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0 | begin
by_cases h_im : s.im = 0,
{ have : s = ↑s.re,
{ conv_lhs { rw ←complex.re_add_im s }, rw [h_im, of_real_zero, zero_mul, add_zero] },
rw [this, Gamma_of_real, of_real_ne_zero],
refine real.Gamma_ne_zero (λ n, _),
specialize hs n,
contrapose! hs,
rwa [this, ←of_real_nat_cast, ←of_real_ne... | theorem | complex.Gamma_ne_zero | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"complex.sin_ne_zero_iff",
"div_ne_zero",
"mul_ne_zero",
"mul_ne_zero_iff",
"real.Gamma_ne_zero",
"zero_mul"
] | The Gamma function does not vanish on `ℂ` (except at non-positive integers, where the function
is mathematically undefined and we set it to `0` by convention). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_eq_zero_iff (s : ℂ) : Gamma s = 0 ↔ ∃ m : ℕ, s = -m | begin
split,
{ contrapose!, exact Gamma_ne_zero },
{ rintro ⟨m, rfl⟩, exact Gamma_neg_nat_eq_zero m },
end | lemma | complex.Gamma_eq_zero_iff | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_ne_zero_of_re_pos {s : ℂ} (hs : 0 < re s) : Gamma s ≠ 0 | begin
refine Gamma_ne_zero (λ m, _),
contrapose! hs,
simpa only [hs, neg_re, ←of_real_nat_cast, of_real_re, neg_nonpos] using nat.cast_nonneg _,
end | lemma | complex.Gamma_ne_zero_of_re_pos | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"nat.cast_nonneg"
] | A weaker, but easier-to-apply, version of `complex.Gamma_ne_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_seq (s : ℝ) (n : ℕ) | (n : ℝ) ^ s * n! / ∏ (j : ℕ) in finset.range (n + 1), (s + j) | def | real.Gamma_seq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"finset.range"
] | The sequence with `n`-th term `n ^ s * n! / (s * (s + 1) * ... * (s + n))`, for real `s`. We
will show that this tends to `Γ(s)` as `n → ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_seq_tendsto_Gamma (s : ℝ) : tendsto (Gamma_seq s) at_top (𝓝 $ Gamma s) | begin
suffices : tendsto (coe ∘ Gamma_seq s : ℕ → ℂ) at_top (𝓝 $ complex.Gamma s),
from (complex.continuous_re.tendsto (complex.Gamma ↑s)).comp this,
convert complex.Gamma_seq_tendsto_Gamma s,
ext1 n,
dsimp only [Gamma_seq, function.comp_app, complex.Gamma_seq],
push_cast,
rw [complex.of_real_cpow n.ca... | lemma | real.Gamma_seq_tendsto_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"complex.Gamma",
"complex.Gamma_seq",
"complex.Gamma_seq_tendsto_Gamma",
"complex.of_real_cpow",
"complex.of_real_nat_cast"
] | Euler's limit formula for the real Gamma function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_mul_Gamma_one_sub (s : ℝ) : Gamma s * Gamma (1 - s) = π / sin (π * s) | begin
simp_rw [←complex.of_real_inj, complex.of_real_div, complex.of_real_sin,
complex.of_real_mul, ←complex.Gamma_of_real, complex.of_real_sub, complex.of_real_one],
exact complex.Gamma_mul_Gamma_one_sub s
end | lemma | real.Gamma_mul_Gamma_one_sub | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"complex.Gamma_mul_Gamma_one_sub",
"complex.of_real_div",
"complex.of_real_mul",
"complex.of_real_one",
"complex.of_real_sin",
"complex.of_real_sub"
] | Euler's reflection formula for the real Gamma function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_div_Gamma_eq_self_mul_one_div_Gamma_add_one (s : ℂ) :
(Gamma s)⁻¹ = s * (Gamma (s + 1))⁻¹ | begin
rcases ne_or_eq s 0 with h | rfl,
{ rw [Gamma_add_one s h, mul_inv, mul_inv_cancel_left₀ h] },
{ rw [zero_add, Gamma_zero, inv_zero, zero_mul] }
end | lemma | complex.one_div_Gamma_eq_self_mul_one_div_Gamma_add_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"inv_zero",
"mul_inv",
"mul_inv_cancel_left₀",
"ne_or_eq",
"zero_mul"
] | A reformulation of the Gamma recurrence relation which is true for `s = 0` as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_one_div_Gamma : differentiable ℂ (λ s : ℂ, (Gamma s)⁻¹) | begin
suffices : ∀ (n : ℕ), ∀ (s : ℂ) (hs : -s.re < n), differentiable_at ℂ (λ u : ℂ, (Gamma u)⁻¹) s,
from λ s, let ⟨n, h⟩ := exists_nat_gt (-s.re) in this n s h,
intro n,
induction n with m hm,
{ intros s hs,
rw [nat.cast_zero, neg_lt_zero] at hs,
suffices : ∀ (m : ℕ), s ≠ -↑m, from (differentiable... | lemma | complex.differentiable_one_div_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"differentiable",
"differentiable_at",
"differentiable_at_const",
"exists_nat_gt",
"nat.cast_nonneg",
"nat.cast_zero"
] | The reciprocal of the Gamma function is differentiable everywhere (including the points where
Gamma itself is not). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * ↑(real.sqrt π) | begin
suffices : (λ z, (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) =
(λ z, (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑(real.sqrt π)),
{ convert congr_arg has_inv.inv (congr_fun this s) using 1,
{ rw [mul_inv, inv_inv, inv_inv] },
{ rw [div_eq_mul_inv, mul_inv, mul_inv, inv_inv, inv_inv, ←cpow_neg, neg_sub] } },
... | theorem | complex.Gamma_mul_Gamma_add_half | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"analytic_on",
"analytic_on.eq_of_frequently_eq",
"differentiable.mul",
"differentiable_at.const_cpow",
"differentiable_at.sub_const",
"differentiable_const",
"differentiable_on",
"differentiable_on.analytic_on",
"div_eq_mul_inv",
"eventually_gt_nhds",
"inv_inv",
"is_open_univ",
"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_mul_Gamma_add_half (s : ℝ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt π | begin
rw [←of_real_inj],
simpa only [←Gamma_of_real, of_real_cpow zero_le_two, of_real_mul, of_real_add, of_real_div,
of_real_bit0, of_real_one, of_real_sub] using complex.Gamma_mul_Gamma_add_half ↑s
end | lemma | real.Gamma_mul_Gamma_add_half | analysis.special_functions.gamma | src/analysis/special_functions/gamma/beta.lean | [
"analysis.convolution",
"analysis.special_functions.trigonometric.euler_sine_prod",
"analysis.special_functions.gamma.bohr_mollerup",
"analysis.analytic.isolated_zeros"
] | [
"Gamma",
"complex.Gamma_mul_Gamma_add_half",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.congr [has_smul 𝕜 β] (hf : convex_on 𝕜 s f) (hfg : eq_on f g s) :
convex_on 𝕜 s g | ⟨hf.1, λ x hx y hy a b ha hb hab,
by simpa only [←hfg hx, ←hfg hy, ←hfg (hf.1 hx hy ha hb hab)] using hf.2 hx hy ha hb hab⟩ | lemma | convex_on.congr | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.congr [has_smul 𝕜 β](hf : concave_on 𝕜 s f) (hfg : eq_on f g s) :
concave_on 𝕜 s g | ⟨hf.1, λ x hx y hy a b ha hb hab,
by simpa only [←hfg hx, ←hfg hy, ←hfg (hf.1 hx hy ha hb hab)] using hf.2 hx hy ha hb hab⟩ | lemma | concave_on.congr | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"concave_on",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.congr [has_smul 𝕜 β] (hf : strict_convex_on 𝕜 s f) (hfg : eq_on f g s) :
strict_convex_on 𝕜 s g | ⟨hf.1, λ x hx y hy hxy a b ha hb hab, by simpa only
[←hfg hx, ←hfg hy, ←hfg (hf.1 hx hy ha.le hb.le hab)] using hf.2 hx hy hxy ha hb hab⟩ | lemma | strict_convex_on.congr | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"has_smul",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.congr [has_smul 𝕜 β] (hf : strict_concave_on 𝕜 s f) (hfg : eq_on f g s) :
strict_concave_on 𝕜 s g | ⟨hf.1, λ x hx y hy hxy a b ha hb hab, by simpa only
[←hfg hx, ←hfg hy, ←hfg (hf.1 hx hy ha.le hb.le hab)] using hf.2 hx hy hxy ha hb hab⟩ | lemma | strict_concave_on.congr | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"has_smul",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.add_const [module 𝕜 β] (hf : convex_on 𝕜 s f) (b : β) :
convex_on 𝕜 s (f + (λ _, b)) | hf.add (convex_on_const _ hf.1) | lemma | convex_on.add_const | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"convex_on_const",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.add_const [module 𝕜 β] (hf : concave_on 𝕜 s f) (b : β) :
concave_on 𝕜 s (f + (λ _, b)) | hf.add (concave_on_const _ hf.1) | lemma | concave_on.add_const | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"concave_on",
"concave_on_const",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on.add_const {γ : Type*} {f : E → γ}
[ordered_cancel_add_comm_monoid γ] [module 𝕜 γ] (hf : strict_convex_on 𝕜 s f) (b : γ) :
strict_convex_on 𝕜 s (f + (λ _, b)) | hf.add_convex_on (convex_on_const _ hf.1) | lemma | strict_convex_on.add_const | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on_const",
"module",
"ordered_cancel_add_comm_monoid",
"strict_convex_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on.add_const {γ : Type*} {f : E → γ}
[ordered_cancel_add_comm_monoid γ] [module 𝕜 γ] (hf : strict_concave_on 𝕜 s f) (b : γ) :
strict_concave_on 𝕜 s (f + (λ _, b)) | hf.add_concave_on (concave_on_const _ hf.1) | lemma | strict_concave_on.add_const | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"concave_on_const",
"module",
"ordered_cancel_add_comm_monoid",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma {s t a b : ℝ}
(hs : 0 < s) (ht : 0 < t) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
Gamma (a * s + b * t) ≤ Gamma s ^ a * Gamma t ^ b | begin
-- We will apply Hölder's inequality, for the conjugate exponents `p = 1 / a`
-- and `q = 1 / b`, to the functions `f a s` and `f b t`, where `f` is as follows:
let f : ℝ → ℝ → ℝ → ℝ := λ c u x, exp (-c * x) * x ^ (c * (u - 1)),
have e : is_conjugate_exponent (1 / a) (1 / b) := real.is_conjugate_exponent_... | lemma | real.Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"continuous.continuous_on",
"continuous_at.continuous_on",
"continuous_id'",
"continuous_on.ae_strongly_measurable",
"ennreal.div_self",
"ennreal.of_real",
"ennreal.of_real_eq_zero",
"ennreal.of_real_ne_top",
"ennreal.to_real_of_real",
"exp",
"measurable_set_Ioi",
"measure_theory.in... | Log-convexity of the Gamma function on the positive reals (stated in multiplicative form),
proved using the Hölder inequality applied to Euler's integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_log_Gamma : convex_on ℝ (Ioi 0) (log ∘ Gamma) | begin
refine convex_on_iff_forall_pos.mpr ⟨convex_Ioi _, λ x hx y hy a b ha hb hab, _⟩,
have : b = 1 - a := by linarith, subst this,
simp_rw [function.comp_app, smul_eq_mul],
rw [←log_rpow (Gamma_pos_of_pos hy), ←log_rpow (Gamma_pos_of_pos hx),
←log_mul
((rpow_pos_of_pos (Gamma_pos_of_pos hx) _).ne') ... | lemma | real.convex_on_log_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"convex_on",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_Gamma : convex_on ℝ (Ioi 0) Gamma | begin
refine ((convex_on_exp.subset (subset_univ _) _).comp convex_on_log_Gamma
(exp_monotone.monotone_on _)).congr (λ x hx, exp_log (Gamma_pos_of_pos hx)),
rw convex_iff_is_preconnected,
refine is_preconnected_Ioi.image _ (λ x hx, continuous_at.continuous_within_at _),
refine (differentiable_at_Gamma (λ m,... | lemma | real.convex_on_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"continuous_at.continuous_within_at",
"continuous_at.log",
"convex_on",
"nat.cast_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_gamma_seq (x : ℝ) (n : ℕ) : ℝ | x * log n + log n! - ∑ (m : ℕ) in finset.range (n + 1), log (x + m) | def | real.bohr_mollerup.log_gamma_seq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"finset.range"
] | The function `n ↦ x log n + log n! - (log x + ... + log (x + n))`, which we will show tends to
`log (Gamma x)` as `n → ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_nat_eq (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : n ≠ 0) :
f n = f 1 + log (n - 1)! | begin
refine nat.le_induction (by simp) (λ m hm IH, _) n (nat.one_le_iff_ne_zero.2 hn),
have A : 0 < (m : ℝ), from nat.cast_pos.2 hm,
simp only [hf_feq A, nat.cast_add, algebra_map.coe_one, nat.add_succ_sub_one, add_zero],
rw [IH, add_assoc, ← log_mul (nat.cast_ne_zero.mpr (nat.factorial_ne_zero _)) A.ne',
... | lemma | real.bohr_mollerup.f_nat_eq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"algebra_map.coe_one",
"mul_comm",
"nat.add_succ_sub_one",
"nat.cast_add",
"nat.cast_mul",
"nat.factorial_ne_zero",
"nat.factorial_succ",
"nat.le_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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