Solving Complex Integration with Residue Theorem

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The discussion focuses on using the residue theorem to evaluate the integral of sinh(ax)/sinh(xπ) from -infinity to +infinity, where a is between -π and π. Participants explore the application of the residue theorem and the necessity of closed paths for integration. One user mentions attempting rectangular trajectories through specific points but encounters difficulties. The conversation emphasizes the importance of selecting appropriate closed paths for successful application of the theorem. Overall, the thread highlights challenges in applying complex integration techniques to this specific problem.
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Homework Statement



use residue theorem to integrate sinh(ax)/sinh(xpi) from -infinity to +infinity, a is between -pi and pi

Homework Equations



residue theorem

3
. The attempt at a solution
i tried rectangular trajection through 0 and ia/pi with the function sinh(az)/sinh(zpi) and some other trajectories but it doesn't really work
 
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The residue theorem applies to integration over closed paths. What closed paths have you used?
 

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