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

$$\int_0^\infty\frac{\cos(ax)}{\sqrt{1+x^2}}dx$$

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.

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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.

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  • $\begingroup$ 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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