Hello,

I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might relate to certain special functions.

Let $$ G(t,x)=\frac{e^{-\frac{x^2}{2t}}}{\sqrt{2\pi t}}. $$ The problem is $$ \int_0^t \frac{G(s,x)}{\sqrt{t-s}} d s =? $$

One integral, that might be useful, is $$ \int_0^t G(s,x) d s = |x|\left(\Phi\left(\frac{|x|}{\sqrt{2t}}\right)-1\right) + 2t G(t,x) $$ where $\Phi(x)$ is the distribution function of the standard normal random variable: $$ \Phi(x) := \int_{-\infty}^x G(1,y) d y. $$

Thank you very much for any hints!

Wish everyone a nice weekend. :-)

Anand