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I was wondering if any one can help me calculate this integral using Mathematica. This is a form of integral representation of Hankel function of first kind:

(-1)^n/(I Pi[]) *
 Integrate[ Exp[-k r (Sinh[t] Cos[tt]] + I Cosh[t] Sin[tt])] Exp[-n t],
     {t,-Infinity, 0, PI[] I, Infinity + PI[] I}, PrincipalValue -> True])
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    $\begingroup$ There are some syntax errors in your code. But apart from that, MA does not know much about the contour integrals. $\endgroup$
    – yarchik
    Commented Nov 29, 2021 at 22:10
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    $\begingroup$ The Integrate function is not going to integrate that along three separate contours as you seem to wish above. Need to integrate over separate contours or parameterize the contours via a Piecewise function if possible. It's not difficult to integrate it numerically or just use the built-in HankelH1 function. $\endgroup$
    – josh
    Commented Nov 29, 2021 at 22:14

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