Everything to Know about Mellin-Barnes Integrals - Part I

Introduction

What is Mellin-Barnes integrals (MBintegrals)



Contour ℒ for MB integrals



Example with Exp for {A,B,C,D}={1,0,0,0}



Example with Exp for {A,B,C,D}={0,1,0,0}



Important literature for beginners



Connection with FourierTransform



Important characteristics for gammas inside of above MBintegral



Operations with

{

α

z

,

z

a

,

z

z

}



Taylor & Fourier series are based on



α

z

,

z

a



zLog[a]







Functions representable through Mellin-Barnes integrals (MeijeG)



Six examples of Mellin-Barnes integrals (MBintegrals)



General MBintegrals defining MeijerG & FoxH functions

MeijerG & generalized MeijerG https://www.wolframalpha.com/input?i=MeijerG



Craziness in definition of MeijerG



Slater’s definition of MeijerG (Lucy Joan Slater (1922 – 2008))



FoxH https://www.wolframalpha.com/input?i=FoxH



Inter relations MeijerG, generalized MeijerG, FoxH & Slater’s version



Evaluation of MBintegrals for MeijerG function

Main formulas for Gamma https://functions.wolfram.com/GammaBetaErf/Gamma/



Representation of MeijerG through residues in left single poles



Representation of MeijerG through residues in right single poles



Representation of MeijerG through residues in left double poles



Example (m,n,p,q}={1,1,1,1} <->

ρ

(1+z)



Differential equation for MeijerG and its solutions



From Marichev’s book 1978

Conversion of Slater’s MB integral by contour ℒ (

ℒ

-∞

∨

ℒ

+∞

∨

{γ-∞,γ+∞}

)



Small addition in lecture

