There is an integration
$\int_{-\infty}^{+\infty}\frac{dx}{(1+x^2)coshx}$
and we set
$g(z)=\frac{dz}{(1+z^2)coshz}$
obviously,
$\lim_{R\rightarrow+\infty}g(z)=0$
but how to find $e^{imz}$ to satisfy Jordan's Lemma,or it need another way to slove?
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Several solutions. Probably, not the simpliest ones... https://math.stackexchange.com/questions/411058/evaluate-the-integral-int-0-infty-frac11x2-coshaxdx?noredirect=1 In fact, a simple rectangular contour in the complex plane can also be used. – Svyatoslav Nov 17 '23 at 01:42