# Infinite integral of Bessel function (2nd kind) product Gaussian

I'm trying to evaluate the following infinite integral:

$$\int_{0}^{\infty} x e^{-\alpha x^2}Y_0(\beta x) \,dx$$

If we apply the general result in Gradshteyn & Ryzhik (eq. 6.631.2 on page 706), with $$\mu=1$$ and $$\nu=0$$ then the result should be

$$\int_{0}^{\infty} x e^{-\alpha x^2}Y_0(\beta x) \,dx=-\alpha^{-\frac{1}{2}}\beta^{-1} \sec\left(-\frac{\pi}{2}\right) \exp\left(-\frac{\beta^2}{8\alpha}\right)\left[-M_{\frac{1}{2},0}\left(\frac{\beta^2}{4\alpha}\right)+W_{\frac{1}{2},0}\left(\frac{\beta^2}{4\alpha}\right)\right]$$

but $$\sec\left(-\frac{\pi}{2}\right)$$ is undefined (diverges).

However, Mathematica gives a finite result in terms of Meijer-G functions which I am not familiar with.

What am I missing?

## 1 Answer

With Maple I have:

$$\int_0^{\infty } x \exp \left(-a x^2\right) Y_0(b x) \, dx=\frac{\sqrt{\frac{b^2}{a}} e^{-\frac{b^2}{4 a}} \text{Ei}\left(\frac{b^2}{4 a}\right)}{2 \pi \sqrt{a} b}$$ where Ei is exponential integral function.

• Thank you very much! Do you know how you would calculate this without using Maple? May 9 '19 at 10:06