In[]:=
$Version
Out[]=
14.2.0 for Microsoft Windows (64-bit) (December 26, 2024)
Using: then:
ArcTan[x]==InverseMellinTransformGamma-sGamma[1-s]Gamma[s],s,1
1-2s
x
1
2
2Gamma-s
3
2
In[]:=
IntegrateGamma-sGamma[1-s]Gamma[s]*,{x,0,Infinity}
1-2s
x
1
2
2Gamma-s
3
2
1
x^2-x+1
Out[]=
If: Csc[πs]Csc[2πs]3Cos+=Gamma+ then:
2
π
4πs
3
3
Sin4πs
3
3(1-2s)
πGamma[1-s]Gamma[s]Gamma[2s]
2
Gamma[1-2s]
3
Gamma[2-2s]Gamma1-4s
3
4s
3
πGamma[1-s]Gamma[s]Gamma[2s]
2
Gamma[1-2s]
Gamma[2-2s]Gamma-Gamma+
1
2
4s
3
1
2
4s
3
In[]:=
1/(2PiI)NContourIntegrateGamma,s∈InfiniteLine,0,{0,1}+1/(2PiI)NContourIntegrate,s∈InfiniteLine,0,{0,1}(*NumericisOK*)
πGamma[1-s]Gamma[s]Gamma[2s]
2
Gamma[1-2s]
3
Gamma[2-2s]Gamma1-4s
3
4s
3
1
4
πGamma[1-s]Gamma[s]Gamma[2s]
2
Gamma[1-2s]
Gamma[2-2s]Gamma-Gamma+
1
2
4s
3
1
2
4s
3
3
16
Out[]=
1.89941+0.
In[]:=
A=System`MeijerGDump`FromSlaterFormGamma,s//ReleaseHold+System`MeijerGDump`FromSlaterForm,s//ReleaseHold
πGamma[1-s]Gamma[s]Gamma[2s]
2
Gamma[1-2s]
3
Gamma[2-2s]Gamma1-4s
3
4s
3
πGamma[1-s]Gamma[s]Gamma[2s]
2
Gamma[1-2s]
Gamma[2-2s]Gamma-Gamma+
1
2
4s
3
1
2
4s
3
Out[]=
MeijerG0,,,,,,,,,,0,,,,,,,-,,,1,
1
6
1
3
1
3
2
3
2
3
5
6
5
6
1
4
3
4
1
6
1
3
1
3
2
3
2
3
5
6
1
6
1
4
3
4
1
3
32
3
3
π
MeijerG0,0,,,,,,,,,,,,,0,0,,,,,,,,-,,,,,1,
1
6
1
3
1
3
1
2
2
3
2
3
5
6
5
6
1
8
3
8
5
8
7
8
1
6
1
3
1
3
1
2
2
3
2
3
5
6
1
6
1
8
3
8
5
8
7
8
1
3
32
3
π
A//FunctionExpand//FullSimplify(*ClosedformisOK*)
Out[]=
2
π
3
3
In[]:=
%//N
Out[]=
1.89941