In[]:=
$Version
Out[]=
14.2.0 for Microsoft Windows (64-bit) (December 26, 2024)
(*--SecondMethodusingNContourIntegralmethod------------------------------------------------------------------------------------------------------------*)
In[]:=
ClearAll["`*"];Remove["`*"]
In[]:=
K1=2;​​K2=2;​​Ω1=1;​​Ω2=1;​​λ=1;​​p=1;​​M=2;​​ρ=1/2;​​​​q=
πExp[1/2(-K1-K2)]
4
Ω1*Ω2
(K1+1)BesselI0,
K1
2
+K1BesselI1,
K1
2
(K2+1)BesselI0,
K2
2
+K2BesselI1,
K2
2
;​​Vr=
1
16Ω1*Ω2
16(K1+1)(K2+1)-
2
π
Exp[-K1-K2]
2
(K1+1)BesselI0,
K1
2
+K1BesselI1,
K1
2

2
(K2+1)BesselI0,
K2
2
+K2BesselI1,
K2
2

;​​a=
M
2
q
Vr
;​​b=
Vr
q
;
In[]:=
w2[γ_]:=
1
8
4-
1
πGamma[1+a]
2
-1-a
b
p
-1-
a
2
1
2
+
-γ/10
10
(-2+ρ)
-1+ρ
1
2
(-1-a)
(-
γ/5
10
λ(-1+ρ)ρΩ1)
*1(2PiI)*NContourIntegrateGamma1+
a
2
-
s
2
Gamma-
1
2
-
a
2
+
s
2
Gamma
1
2
-
a
2
+
s
2
Gamma[s]*
2p
Ω1*ρ
γ
10
10
*
2
b
λ2(2-ρ)+
γ
10
10
*(1-ρ)
^-s,s∈InfiniteLine
1
2
-2-1-
a
2
+Max0,2-
1
2
+
a
2
,2
1
2
+
a
2
,0,{0,1};
In[]:=
w2[1]
Out[]=
0.38912+0.
In[]:=
data2=Table[{γ,w2[γ]},{γ,1,50,1}];
In[]:=
ListLinePlot[data2//Re]
Out[]=
(*--Extras:----------------------------------------------------------------------------------------------------------*)
(*MellinTransformofFoxHfunction:*)
In[]:=
ClearAll["`*"];Remove["`*"];
In[]:=
F=FoxH-
a
2
,12,{},{0,1},
1-a
2
,12,-
a+1
2
,12,{},x;​​p=Length[Flatten[F[[1]],1]];​​q=Length[Flatten[F[[2]],1]];​​m=Length[F[[2,1]]];​​n=Length[F[[1,1]]];
Product[Gamma[b[j]+B[j]*s],{j,1,m}]Product[Gamma[1-a[i]-A[i]*s],{i,1,n}]
Product[Gamma[a[i]+A[i]*s],{i,n+1,p}]*Product[Gamma[1-b[j]-B[j]*s],{j,m+1,q}]
*(Coefficient[F[[3]],x])^-s/.Flatten[{Table[a[j]->Transpose[Flatten[F[[1]],1]][[1,j]],{j,1,Length[Transpose[Flatten[F[[1]],1]][[1]]]}],Table[A[j]->Transpose[Flatten[F[[1]],1]][[2,j]],{j,1,Length[Transpose[Flatten[F[[1]],1]][[2]]]}],Table[b[j]->Transpose[Flatten[F[[2]],1]][[1,j]],{j,1,Length[Transpose[Flatten[F[[2]],1]][[1]]]}],Table[B[j]->Transpose[Flatten[F[[2]],1]][[2,j]],{j,1,Length[Transpose[Flatten[F[[2]],1]][[2]]]}]}]//Quiet
Out[]=
Gamma1+
a
2
-
s
2
Gamma
1
2
(-1-a)+
s
2
Gamma
1-a
2
+
s
2
Gamma[s]
(*FindingCorrectContourforFoxHfunction:*)
In[]:=
candidatePole[lin_]:=Module[{a,b},a=Coefficient[lin,s];​​b=lin/.s->0;​​If[a>0,-b/a,If[a<0,-b/a,0]]]​​​​leftCandidates[expr_,s_Symbol]:=Module[{gammas,poles},gammas=Cases[expr,Gamma[lin_]:>lin,Infinity];​​poles=Table[With[{a=Coefficient[lin,s],b=lin/.s->0},If[a>0,-b/a,Nothing]],{lin,gammas}];​​poles]​​​​rightCandidates[expr_,s_Symbol]:=Module[{gammas,poles},gammas=Cases[expr,Gamma[lin_]:>lin,Infinity];​​poles=Table[With[{a=Coefficient[lin,s],b=lin/.s->0},If[a<0,-b/a,Nothing]],{lin,gammas}];​​poles]​​​​getContour[expr_,s_Symbol]:=Module[{left,right,cMin,cMax,gamma},left=leftCandidates[expr,s];​​right=rightCandidates[expr,s];​​cMin=If[left==={},-Infinity,Max[left]];​​cMax=If[right==={},Infinity,Min[right]];​​gamma=(cMin+cMax)/2;​​InfiniteLine[{gamma,0},{0,1}]]​​​​expr1=Gamma1+
a
2
-
s
2
Gamma-
1
2
-
a
2
+
s
2
Gamma
1
2
-
a
2
+
s
2
Gamma[s];​​contour1=getContour[expr1,s]
Out[]=
InfiniteLine
1
2
-2-1-
a
2
+Max0,2-
1
2
+
a
2
,2
1
2
+
a
2
,0,{0,1}