How to evaluate $int_{0}^{infty} x^{nu} frac{e^{-sqrt{x^2+a^2}}}{sqrt{x^2+a^2}} , dx$?

Question

$$int_{0}^{infty} x^{nu} frac{e^{-sqrt{x^2+a^2}}}{sqrt{x^2+a^2}} , dx$$
Is it possible to calculate this for $a>0$ and $nu=0, 2$ ?
I think the result seems to include exponential integral function, but I failed to find the answer from the integration table.
I would be very grateful if you could share some of the good integration skills, ideas, or any advice.

Z Ahmed · Answer

Let $x=a sinh t$, then
$$I_1=int_{0}^{infty} frac{e^{-sqrt{x^2+a^2}}}{sqrt{x^2+a^2}} dx= int_{0}^{infty} e^{-a cosh t} dt=K_0(a),$$
where $K_{nu}(x)$ is cylindrical modified Bessel function of order $nu$.
Next, $$I_1=int_{0}^{infty} x^2 frac{e^{-sqrt{x^2+a^2}}}{sqrt{x^2+a^2}} dx=frac{a^2}{2}int_{0}^{infty} (cosh 2t-1)~ e^{-acosh t} dt=frac{a^2}{2} [K_2(a)-K_0(a)]$$
See for $K_{nu}(z):$
https://en.wikipedia.org/wiki/Bessel_function

How to evaluate $int_{0}^{infty} x^{nu} frac{e^{-sqrt{x^2+a^2}}}{sqrt{x^2+a^2}} , dx$?

One Answer

Add your own answers!

Ask a Question