Can Mathematica verify the Cahen-Mellin integral?

When I try

Integrate[Gamma[1 + I s]/Zeta[3]^(1 + I s), {s, -Infinity, Infinity}]

I just get the input back. Thanks for any suggestion!


Numerically it is easy to get agreement:

NIntegrate[Gamma[1 + I s]/Zeta[3]^(1 + I s), {s, -Infinity, Infinity}]

1.88857 + 2.26472*10^-13 I

2 \[Pi] Exp[-Zeta[3]] // N


But I wonder whether Mathematica more generally is capable of doing such integrals analytically.

  • 1
    $\begingroup$ So you just want to use Integrate rather than InverseMellinTransform? If so, slightly related: mathematica.stackexchange.com/q/81154/1871 $\endgroup$
    – xzczd
    Mar 15, 2017 at 4:33
  • $\begingroup$ Its nice to know that Mathematica has explicit mellin transform functions! Come to think of it, mathematica can't do Integrate[Exp[I t q],{t,-Infinity,Infinity}] either, but FourierTransform[1,t,q] works perfectly fine. $\endgroup$
    – Kagaratsch
    Mar 15, 2017 at 19:33


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.