# The Gamma function #

This file defines the Γ function (of a real or complex variable s). We define this by Euler's integral Γ(s) = ∫ x in Ioi 0, exp (-x) * x ^ (s - 1) in the range where this integral converges (i.e., for 0 < s in the real case, and 0 < re s in the complex case).

We show that this integral satisfies Γ(1) = 1 and Γ(s + 1) = s * Γ(s); hence we can define Γ(s) for all s as the unique function satisfying this recurrence and agreeing with Euler's integral in the convergence range. (If s = -n for n ∈ ℕ, then the function is undefined, and we set it to be 0 by convention.)

## Gamma function: main statements (complex case) #

• Complex.Gamma: the Γ function (of a complex variable).
• Complex.Gamma_eq_integral: for 0 < re s, Γ(s) agrees with Euler's integral.
• Complex.Gamma_add_one: for all s : ℂ with s ≠ 0, we have Γ (s + 1) = s Γ(s).
• Complex.Gamma_nat_eq_factorial: for all n : ℕ we have Γ (n + 1) = n!.
• Complex.differentiableAt_Gamma: Γ is complex-differentiable at all s : ℂ with s ∉ {-n : n ∈ ℕ}.

## Gamma function: main statements (real case) #

• Real.Gamma: the Γ function (of a real variable).
• Real counterparts of all the properties of the complex Gamma function listed above: Real.Gamma_eq_integral, Real.Gamma_add_one, Real.Gamma_nat_eq_factorial, Real.differentiableAt_Gamma.

## Tags #

Gamma

theorem Real.Gamma_integrand_isLittleO (s : ) :
(fun (x : ) => Real.exp (-x) * x ^ s) =o[Filter.atTop] fun (x : ) => Real.exp (-(1 / 2) * x)

Asymptotic bound for the Γ function integrand.

theorem Real.GammaIntegral_convergent {s : } (h : 0 < s) :
MeasureTheory.IntegrableOn (fun (x : ) => Real.exp (-x) * x ^ (s - 1)) () MeasureTheory.volume

The Euler integral for the Γ function converges for positive real s.

theorem Complex.GammaIntegral_convergent {s : } (hs : 0 < s.re) :
MeasureTheory.IntegrableOn (fun (x : ) => (Real.exp (-x)) * x ^ (s - 1)) () MeasureTheory.volume

The integral defining the Γ function converges for complex s with 0 < re s.

This is proved by reduction to the real case.

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.GammaIntegral_convergent for a proof of the convergence of the integral for 0 < re s.

Equations
Instances For
theorem Complex.GammaIntegral_ofReal (s : ) :
= (∫ (x : ) in , Real.exp (-x) * x ^ (s - 1))

Now we establish the recurrence relation Γ(s + 1) = s * Γ(s) using integration by parts.

def Complex.partialGamma (s : ) (X : ) :

The indefinite version of the Γ function, Γ(s, X) = ∫ x ∈ 0..X, exp(-x) x ^ (s - 1).

Equations
Instances For
theorem Complex.tendsto_partialGamma {s : } (hs : 0 < s.re) :
Filter.Tendsto (fun (X : ) => ) Filter.atTop ()
theorem Complex.partialGamma_add_one {s : } (hs : 0 < s.re) {X : } (hX : 0 X) :
Complex.partialGamma (s + 1) X = - (Real.exp (-X)) * X ^ s

The recurrence relation for the indefinite version of the Γ function.

theorem Complex.GammaIntegral_add_one {s : } (hs : 0 < s.re) :

The recurrence relation for the Γ integral.

Now we define Γ(s) on the whole complex plane, by recursion.

noncomputable def Complex.GammaAux :

The nth function in this family is Γ(s) if -n < s.re, and junk otherwise.

Equations
Instances For
theorem Complex.GammaAux_recurrence1 (s : ) (n : ) (h1 : -s.re < n) :
= Complex.GammaAux n (s + 1) / s
theorem Complex.GammaAux_recurrence2 (s : ) (n : ) (h1 : -s.re < n) :
@[irreducible]
def Complex.Gamma (s : ) :

The Γ function (of a complex variable s).

Equations
Instances For
theorem Complex.Gamma_eq_GammaAux (s : ) (n : ) (h1 : -s.re < n) :
theorem Complex.Gamma_add_one (s : ) (h2 : s 0) :

The recurrence relation for the Γ function.

theorem Complex.Gamma_eq_integral {s : } (hs : 0 < s.re) :
@[simp]
@[simp]

At 0 the Gamma function is undefined; by convention we assign it the value 0.

At -n for n ∈ ℕ, the Gamma function is undefined; by convention we assign it the value 0.

theorem Complex.Gamma_conj (s : ) :
Complex.Gamma (() s) = () ()
theorem Complex.integral_cpow_mul_exp_neg_mul_Ioi {a : } {r : } (ha : 0 < a.re) (hr : 0 < r) :
∫ (t : ) in , t ^ (a - 1) * Complex.exp (-(r * t)) = (1 / r) ^ a *

Expresses the integral over Ioi 0 of t ^ (a - 1) * exp (-(r * t)) in terms of the Gamma function, for complex a.

Now check that the Γ function is differentiable, wherever this makes sense.

Rewrite the Gamma integral as an example of a Mellin transform.

theorem Complex.hasDerivAt_GammaIntegral {s : } (hs : 0 < s.re) :
HasDerivAt Complex.GammaIntegral (∫ (t : ) in , t ^ (s - 1) * (() * (Real.exp (-t)))) s

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).

theorem Complex.differentiableAt_GammaAux (s : ) (n : ) (h1 : 1 - s.re < n) (h2 : ∀ (m : ), s -m) :
theorem Complex.differentiableAt_Gamma (s : ) (hs : ∀ (m : ), s -m) :

At s = 0, the Gamma function has a simple pole with residue 1.

def Real.Gamma (s : ) :

The Γ function (of a real variable s).

Equations
• = ().re
Instances For
theorem Real.Gamma_eq_integral {s : } (hs : 0 < s) :
= ∫ (x : ) in , Real.exp (-x) * x ^ (s - 1)
theorem Real.Gamma_add_one {s : } (hs : s 0) :
Real.Gamma (s + 1) = s *
@[simp]
theorem Real.Gamma_one :
= 1
theorem Complex.Gamma_ofReal (s : ) :
= ()
theorem Real.Gamma_nat_eq_factorial (n : ) :
Real.Gamma (n + 1) = ()
@[simp]
@[simp]
theorem Real.Gamma_zero :
= 0

At 0 the Gamma function is undefined; by convention we assign it the value 0.

At -n for n ∈ ℕ, the Gamma function is undefined; by convention we assign it the value 0.

theorem Real.Gamma_pos_of_pos {s : } (hs : 0 < s) :
0 <
theorem Real.Gamma_nonneg_of_nonneg {s : } (hs : 0 s) :
0
theorem Real.integral_rpow_mul_exp_neg_mul_Ioi {a : } {r : } (ha : 0 < a) (hr : 0 < r) :
∫ (t : ) in , t ^ (a - 1) * Real.exp (-(r * t)) = (1 / r) ^ a *

Expresses the integral over Ioi 0 of t ^ (a - 1) * exp (-(r * t)), for positive real r, in terms of the Gamma function.

The positivity extension which identifies expressions of the form Gamma a.

Instances For
theorem Real.Gamma_ne_zero {s : } (hs : ∀ (m : ), s -m) :
0

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).

theorem Real.Gamma_eq_zero_iff (s : ) :
= 0 ∃ (m : ), s = -m
theorem Real.differentiableAt_Gamma {s : } (hs : ∀ (m : ), s -m) :