Everything to Know about Mellin-Barnes Integrals - Part 2

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 (1983)

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

ℒ

-∞

∨

ℒ

+∞

∨

{γ-∞,γ+∞}

)



Small addition in lecture



Mellin-Barnes integrals by Slater’s approach (part 2)

What is Mellin-Barnes integrals (MBintegrals, more)



MeijerG & SlaterG functions as cases of FoxH & SlaterH functions



Six examples of Mellin-Barnes integrals (MBintegrals)



Six examples of Mellin-Barnes integrals (MBintegrals)



Important characteristics for gammas inside of SlaterH MBintegral



Asymptotic of “gamma product”



A

∏

k=1

Gamma[

a

k

+

α

k

s]

B

∏

k=1

Gamma[

b

k

-

β

k

s]



C

∏

k=1

Gamma[

c

k

+

γ

k

s]

D

∏

k=1

Gamma[

d

k

-

δ

k

s]

-s

z



Convolution & Parseval equality for MellinTransform



Example of using Parseval equality for MellinTransform



Integral from one FoxH (Oleg)



General integral from product of two SlaterH functions



Integral representations of SlaterH function



Laplace transform of SlaterH function



Laplace transform of MittagLefflerE function



Evaluation SlaterH function through analogs of PFQ functions



MeijerG & FoxH (additions 1)

Derivative of FoxH



BesselMaitland



GeneralizedStruveH



MLefflerE



FoxH cases



Cases of FoxH



Main terms of asymptotics for FoxH at zero



Main terms of asymptotics for FoxH at zero for m0 ∧ real z>0



Main terms of asymptotics for FoxH at ∞



Main terms of asymptotics for FoxH at ∞ for n0 ∧ real z>0



Main terms of asymptotics for MittagLefflerE[α,z]



Integrals (cases)



Integral from one FoxH



Integral from two FoxH



FoxH (Oleg)



MeijerG & FoxH (additions 2)

Expansions for qp+1 ||| Expansions for

qp+1



Expansions for qp+2 ||| Expansions for

qp+2



Expansions for q>p+2 ||| Expansions for

q>p+2



General formulas of asymptotic series expansions



Programs



Main terms for AsymptoticMeijerGSeries[Power]



Main terms for AsymptoticMeijerGSeries[Exp] q-p1



Main terms for AsymptoticMeijerGSeries[Trig] q-p2



Main terms for AsymptoticMeijerGSeries[Hyp] q-p>2



Conditions for FoxH (preliminaries)



FoxH of two variables



Integral from 2 FoxH case 1-2 gen



Integral from 2 FoxH case 1



Integral from 2 FoxH case 2



Verification of Integral from 2 FoxH case 1



Verification of Integral from 2 FoxH case 2

