CITE THIS NOTEBOOK: Wolfram R&D LIVE: Everything to know about Mellin-Barnes Integrals - PartII by Oleg Marichev. Wolfram Community MAR 28 2023.

Introduction

What is Mellin-Barnes integrals (MBintegrals)

Pincherle S . (1888), Mellin H . (1910), Barnes E . W . (1908), Dixon A . L ., Ferrar W . L . (1936)
In Slater' type notations the MBintegral is (
s->
1
s
sometimes)
MBintegral=
1
2π
γ+∞
∫
γ-∞
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
s​​(*
1
2π
γ+∞
∫
γ-∞
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
sSimmetry*)
α
k
>0∧
β
k
>0∧
γ
k
>0∧
δ
k
>0∧Re[
a
k
+
α
k
s]>0∧Re[
b
k
-
β
k
s]>0∧γRe[s]
Music, easy to remember: {A, B, C, D}, {a, b, c, d}, {α, β, γ, δ}
A+D-B-C>or<or==0(*direction'schoice
z
z
*)
A+B-C-D>or==0(*sectorexisting
z

*)
ℒ={γ-∞,γ+∞}(*ℒverticalorleftloop
ℒ
-∞
orrightloop
ℒ
∞
*)
In[]:=

s
z
,
-s
1
z
/.{z->-2,s->Random[Complex]}
In[]:=
TableWith{z=Random[]Exp[PiIkk/16],s=Random[Complex]},
s
z

-s
1
z
,{kk,0,34}//Chop
See
https://functions.wolfram.com/ElementaryFunctions/Power/16/05/01/0018/
In[]:=
PowerExpand[
b
(
a
z
)
,Assumptions->True]
2bπFloor
1
2
-
Im[aLog[z]]
2π


ab
z
1
2π
γ+∞
∫
γ-∞
-
-s
z
2
Gamma[-s]
Gamma[1+s]
Gamma[1-s]
s/;-1<γ<0
In[]:=
ResourceFunction["MeijerGForm"][Log[1+z],z]
Out[]=
MeijerG[{{1,1},{}},{{1},{0}},z]
In[]:=
System`MeijerGDump`SlaterForm[MeijerG[{{1,1},{}},{{1},{0}},z],s]
Out[]=
-s
z
2
Gamma[-s]
Gamma[1+s]
Gamma[1-s]
Gamma[1+s]==ComplexInfinity/;s=-1,-2,-3,…
Re[s+1]>0
In[]:=
-s
z
Gamma[s]Gamma[-s]==-
-s
z
2
Gamma[-s]
Gamma[1+s]
Gamma[1-s]
//FullSimplify
Out[]=
True
-1<γ<0

Contour ℒ for MB integrals

In[]:=
Import["C:\\Users\\oleg\\Desktop\\DOC004.jpg"]
Out[]=

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

Poles of gamma function :
Gamma[s]==ComplexInfinity/;a->-k&&k∈Integers&&k>=0
In[]:=
Series[Gamma[s],{s,-k,1},Assumptions->{k∈Integers&&k>=0}]
1
2π
γ+∞
∫
γ-∞
Gamma[s]
-s
z
s
-z

1
2π
γ+∞
∫
γ-∞
Gamma[
a
1
+
α
1
s]
-s
z
s
-
1
α
1
z

a
1
α
1
z
α
1
In[]:=
Solve[
a
1
+
α
1
s==-k,s]
plus Residue :
In[]:=
ResidueGamma[
a
1
+
α
1
s]
-s
z
,s,
-k-
a
1
α
1
,Assumptions->{k∈Integers&&k>=0}
In[]:=
Sum[%,{k,0,∞}]
Formally with 3 steps for left SIMPLE poles (for right poles we need to multiply on -1):
In[]:=
Gamma1[
a
1
+
α
1
s]
-s
z
/.s->
-k-
a
1
α
1

In[]:=
%/.Gamma1[-k]->
k
(-1)
α
1
k!
In[]:=
Sum[%,{k,0,∞}]

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

Important literature for beginners


Connection with FourierTransform

https://functions.wolfram.com/GeneralIdentities/11/
FourierTransform[f[t],t,z]
1
2π
∞
∫
-∞
f[t]
tz

t
FourierTransform
∞
∫
-∞
f[t-τ]g[τ]τ,t,z
2π
FourierTransform[f[t],t,z]FourierTransform[g[t],t,z]
InverseFourierTransform[f[t],t,z]
1
2π
∞
∫
-∞
f[t]
-tz

t
FourierTransform[f[t],t,z]
1
2π
MellinTransform[f[-Log[t]],t,-z]
FourierTransform[f[t],t,z]
2π
γz

InverseMellinTransform[f[(s-γ)],s,
z

]
MellinTransform[f[t],t,z]
∞
∫
0
f[t]
z-1
t
t
MellinTransform
a
t
∞
∫
0
b-1
τ
f[
c
τ
]g[t
e
τ
]τ,t,z
1
Abs[c]
MellinTransformf[t],t,
b-e(a+z)
c
MellinTransform[g[t],t,a+z]/;c≠0∧c∈Reals∧e∈Reals
∞
∫
0
f[τ]g[τ]τ
1
2π
γ+∞
∫
γ-∞
MellinTransform[f[t],t,s]MellinTransform[g[t],t,1-s]s
InverseMellinTransform[f[s],{s,γ},t]
1
2π
γ+∞
∫
γ-∞
f[s]
-s
t
s
MellinTransform[f[t],t,z]
2π
FourierTransform[f[
-t

],t,z]
InverseMellinTransform[f[s],{s,γ},t]
-γ
t
2π
FourierTransform[f[γ-s],s,Log[t]]
MellinTransform[f[τ],τ,z]
∞
∫
0
f[τ]
z-1
τ
τ
2π
FourierTransform[f[
-t

],t,z]
2π
1
2π
∞
∫
-∞
f[
-t

]
t(z)

t(*/.t->Log
1
τ
*)
∞
∫
0
f[τ]
z-1
τ
τ
In[]:=
Solve[
-t

τ,t]
Finally

General MBintegrals defining MeijerG & FoxH functions

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

Meijer C . S . (1936, 1941, 1946)

From Marichev’s book 1978 (1983)

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

What is Mellin-Barnes integrals (MBintegrals, more)

Pincherle S . (1888), Mellin H . (1910), Barnes E . W . (1908), Dixon A . L ., Ferrar W . L . (1936)
Music, easy to remember: {A, B, C, D}, {a, b, c, d}, {α, β, γ, δ}
SlaterH is the maximally convenient for users equivalent of FoxH function, defined in 1961 (Fox, C. “The G and H-Functions as Symmetrical Fourier Kernels.” Trans. Amer. Math. Soc. 98, 395-429, 1961.)
and implemented to Mathematica by Tigran Ishkhanyan (https://www.youtube.com/watch?v=jFspWjZm_pk)

MeijerG & SlaterG functions as cases of FoxH & SlaterH functions

Lucy J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, London-New York, 1966.
SlaterH -> FoxH
FoxH -> SlaterH

Laplace transform of MittagLefflerE function