Evaluate $\int_{-\infty}^{+\infty}\frac{1}{\sqrt{x^2+1}}\cdot\frac{1}{e^{\beta\sqrt{x^2+1}}+1}dx$ using contour integration with branch cuts

Asked Feb 25 '23 at 16:14

Active Feb 25 '23 at 16:14

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I want to calculate the integral $$\int_{-\infty}^{+\infty}\frac{1}{\sqrt{x^2+1}}\cdot\frac{1}{e^{\beta\sqrt{x^2+1}}+1}dx$$ for $\beta>0$ with the residue theorem. I tried to follow the steps in this example, using the same contour, but I couldn't get any meaningful results.

asked Feb 25 '23 at 16:14

user1153717

1

Not to be pessimistic, but there might not be any simple closed form expression for the integral. It is possible that the two integrals are different enough that the methods used to evaluate the other integral cannot be applied here. – cpiegore Feb 25 '23 at 16:28

Evaluate $\int_{-\infty}^{+\infty}\frac{1}{\sqrt{x^2+1}}\cdot\frac{1}{e^{\beta\sqrt{x^2+1}}+1}dx$ using contour integration with branch cuts

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