EMILY GULLERUD, michael vaughan
University of Wisconsin - Eau Claire
May 10, 2017
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 1 / 15 What is the Gamma Function? Motivation
The gamma function (Γ) is an extension of the factorial function to − C \ Z ∪ 0. It is a solution to the interpolation problem of connecting the discrete points of the factorial function.
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 2 / 15 What is the Gamma Function? Definitions
Factorial function definition + If n ∈ Z , then Γ(n) = (n − 1)!
Improper integral definition R ∞ z−1 −x If z ∈ {x + iy | x > 0}, then Γ(z) = 0 x e dx.
Analytic continuation is required to extend the integral definition to − z ∈ {x + iy | y 6= 0 when x ∈ Z ∪ 0}. This function has simple poles at all the non-positive integers.
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 3 / 15 What is the Gamma Function? More Definitions
Definition as an infinite product ∞ 1 z 1 Y 1 + Γ(z) = n z 1 + z n=1 n
Weierstrass’s definition ∞ e−γz Y z −1 Γ(z) = 1 + ez/n z n n=1
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 4 / 15 Properties Identities and Formulas
Requirements for Γ to be an extension of the factorial function: Γ(1) = 1
zΓ(z) = Γ(z + 1) for Re(z) > 0 Property of conjugation: Γ(z) = Γ(z) Complement Formula: π Γ(z)Γ(1 − z) = for z ∈/ sin(πz) Z Duplication Formula: 1 1−2z √ Γ(z)Γ(z + 2 ) = Γ(2z) · 2 π
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 5 / 15 Properties Proof of zΓ(z) = Γ(z + 1)
Let α ∈ C such that Re(α) > 0 Z ∞ Γ(α) = xα−1e−x dx 0 ∞ Z ∞ α−1 −x −x α−2 = x (−e ) − −e (α − 1)x dx 0 0 α−1 Z ∞ x −x α−2 = − lim x + e (α − 1)x dx x→∞ e 0 Z ∞ = 0 + (α − 1) e−x xα−1−1dx 0 Γ(α) = (α − 1)Γ(α − 1)
Take z = α − 1. Γ(z + 1) = zΓ(z)
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 6 / 15 Properties Values of the Gamma Function
3 4√ Γ − = π 2 3 1 √ Γ − = −2 π 2 Γ(1) = 1 1 √ Γ(2) = 1 Γ = π 2 Γ(3) = 2 3 1√ Γ = π Γ(4) = 6 2 2 Γ(5) = 24 5 3√ Γ = π 2 4 7 15√ Γ = π 2 8
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 7 / 15 Properties √ 1 π Example: Proving 2 ! = 2
1 3 ! = Γ 2 2 1 1 = Γ 2 2 1/2 Z ∞ Let u = t 1 −1/2 −t 1 −1/2 = t e dt Then du = 2 t dt 2 0 ∞ Z 2 = e−u √0 π = 2
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 8 / 15 Applications and Connections to Other Functions The log-gamma function
The Gamma function grows rapidly, so taking the natural logarithm yields a function which grows much more slowly:
ln Γ(z) = ln Γ(z + 1) − ln z
This function is used in many computing environments and in the context of wave propogation.
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 9 / 15 Applications and Connections to Other Functions Derivative of the log-gamma function: Digamma function
The Digamma function is defined to be the logarithmic derivative of the Gamma function: d Γ0 (z) ψ(z) = ln (Γ(z)) = dz Γ(z) A general form of the Digamma function: the Polygamma function defined to be the (m + 1)th logarithmic derivative of the Gamma function: dm ψ(m)(z) = ln (Γ(z)) dzm Notice that ψ(0)(z) = ψ(z) The Polygamma function is meromorphic on C (holomorphic on − C \ Z ∪ 0 The nonpositive integers have poles of order m + 1
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 10 / 15 Applications and Connections to Other Functions Digamma and Polygamma functions
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 11 / 15 Applications and Connections to Other Functions Digamma and Polygamma functions
Finding a property of the Digamma function: Recall that Γ(z + 1) = zΓ(z) Take the derivative:
0 0 Γ (z + 1) = zΓ (z) + Γ(z)
Divide by Γ(z + 1) = zΓ(z):
Γ0 (z + 1) Γ0 (z) 1 = + Γ(z + 1) Γ(z) z
Substitute in ψ function: 1 ψ(z + 1) = ψ(z) + z
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 12 / 15 Applications and Connections to Other Functions Incomplete Gamma Functions
upper incomplete gamma function R ∞ s−1 −t Γ(s, x) = x x e dt
lower incomplete gamma function R x s−1 −t γ(s, x) = 0 x e dt
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 13 / 15 Applications and Connections to Other Functions Pi and Beta Functions
Definition of the Beta Function R 1 x−1 y−1 B(x, y) = 0 t (1 − t) dt for Re(x), Re(y) > 0 Beta function in terms of the Gamma function: Γ(x)Γ(y) B(x, y) = Γ(x + y)
Definition of the Pi Function Π(z) = Γ(z + 1) = zΓ(z)
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 14 / 15 Applications and Connections to Other Functions Riemann Zeta Function
The functional equation: πs ζ(s) = 2s πs−1 sin Γ(1 − s)ζ(1 − s) 2 The functional equation in another form: − s s − 1−s 1 − s π 2 Γ ζ(s) = π 2 Γ ζ(1 − s) 2 2
Another relation: Z ∞ uz−1 ζ(z)Γ(z) = u du for Re(z) > 1 0 e − 1
EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 15 / 15