13. Gamma Function 1.Definitions, Properties 2.Digamma & Polygamma Functions 3.The Beta Function 4.Sterling’s Series 5.Riemann Zeta Function 6.Other Related.

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Presentation transcript:

13. Gamma Function 1.Definitions, Properties 2.Digamma & Polygamma Functions 3.The Beta Function 4.Sterling’s Series 5.Riemann Zeta Function 6.Other Related Functions

Peculiarities: 1. Do not satisfy any differential equation with rational coefficients. 2. Not a hypergeometric nor a confluent hypergeometric function. Common occurence: In expansion coefficients.

13.1.Definitions, Properties Definition, infinite limit (Euler) version : 

Definition: Definite Integral Definition, definite integral (Euler) version : , else singular at t = 0.

Equivalence of the Limit & Integral Definitions Consider  

Definition: Infinite Product (Weierstrass Form) Definition, Infinite Product (Weierstrass) version : Euler-Mascheroni constant Proof : 

Functional Relations Reflection formula : ( about z = ½ ) Proof : Let 

f (z) has pole of order m at z 0 : For z  integers, set branch cut ( for v z ) = + x-axis :   

Legendre’s Duplication Formula General proof in §13.3. Proof for z = n = 1, 2, 3, …. :  ( Case z = 0 is proved by inspection. )

Analytic Properties Weierstrass form :  has simple zeros at z   n, no poles.   (z) has simple poles at z   n, no zeros.  changes sign at z   n. Minimum of  for x > 0 is Mathematica

Residues at z   n Residue at simple pole z   n is n + 1 times : 

Schlaefli Integral Schlaefli integral : Proof : C 1 is an open contour. ( e  t   for Re t   . Branch-cut. )  if >  1 

For Re <  1, I A, I B, & I D are all singular. However, remains finite. ( integrand regular everywhere on C ) Factorial function :  (z) is the Gauss’ notation For Re >  1, I D = 0  reproduces the integral represention. where  is valid for all.

Example Maxwell-Boltzmann Distribution Classical statistics (for distinguishable particles) : Probability of state of energy E being occupied is Maxwell-Boltzmann distribution Partition function  Average energy : g(E) = density of states Ideal gas : gamma distribution

13.2.Digamma & Polygamma Functions Digamma function :    50 digits z = integer : Mathematica

Polygamma Function Polygamma Function :   = Reimann zeta function Mathematica

Maclaurin Expansion of ln   Converges for Stirling’s series ( § 13.4 ) has a b etter convergence.

Series Summation Example Catalan’s Constant Dirichlet series : Catalan’s Constant : 20 digits Mathematica

13.3.The Beta Function Beta Function :  

Alternate Forms : Definite Integrals    To be used in integral rep. of Bessel (Ex ) & hypergeometric (Ex ) functions

Derivation: Legendre Duplication Formula  