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

Mathematics Asked by Ui-Jin Kwon on November 19, 2021

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

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.

One 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

Answered by Z Ahmed on November 19, 2021

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