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?


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.

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

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