I have no idea to evaluate the following integral and I am in trouble:


since the square root prevents me from using an ordinary way to evaluate real integrals with residue theorem. Does anyone know how to solve the above integral?

Thanks in advance.


If you have a look here, you will see that $$\int_0^\infty \frac{\cos(ax)}{\sqrt{1+x^2}}dx=K_0(|a|)$$ where appears the modified Bessel function of the second kind.

  • $\begingroup$ As with any special-functions answer, it's worth bearing in mind that any derivation of this result will depend on how $K_0$ is defined. (Indeed, the above integral is itself a possible definition.) $\endgroup$ – Semiclassical Jul 26 '17 at 3:35
  • $\begingroup$ @Semiclassical. For sure, you are correct ! $\endgroup$ – Claude Leibovici Jul 26 '17 at 3:37

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