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



A definite integral
Happy Birthday to Mathoverflow. Wish it flourish and thank many warmhearted people here for their helps! :-)


Here is one solution. Let 


$$
G\_\sigma(t,x)=\frac{\exp(-\frac{x^2}{2\sigma t})}{\sqrt{2\pi \sigma t}}
$$


Clearly, 


$$
\int\_0^t \frac{G\_\sigma(t-s,x)}{\sqrt{s}} d s = 
\int\_0^t \frac{e^{-\frac{x^2}{2\nu s}}}{\sqrt{2\pi s (t-s)}} d s\;.
$$


We assume that $x\ne 0$. Then by change of variable


$$
s\rightarrow u=\frac{x^2}{2\sigma s}-\frac{x^2}{2\sigma t}, \quad s=
\frac{t x^2}{2\sigma t u+x^2},
$$


the integral becomes


$$
\frac{|x|e^{-\frac{x^2}{2\sigma t}}}{2\sigma \sqrt{\pi t}}\int\_{0}^\infty
\frac{e^{-u}}{\sqrt{u}\left(u+\frac{x^2}{2\sigma t}\right)} d u =
\sqrt{\frac{\pi}{2\sigma}}\left(1-\Phi\left(\frac{|x|}{\sqrt{2\sigma t}}
\right)\right)\:,
$$


where we have applied the integral [(7.4.9) in P. 302](http://en.wikipedia.org/wiki/Abramowitz_and_Stegun)


$$
\int\_0^\infty\frac{e^{-at}}{\sqrt{t}(t+z)} d t =
\frac{2\pi}{\sqrt{z}}e^{a z} \left(1-\Phi(\sqrt{az})\right),\quad
Re(a)>0, z\ne 0, |\arg z|<\pi\:
$$


with $a=1$ and $z=\frac{x^2}{2\sigma t}$, where we have used the fact that $\text{Erfc}(x)=2(1-\Phi(x))$. This then proves


$$
\boxed{
\int\_0^t \frac{G\_\sigma(t-s,x)}{\sqrt{s}} d s =
\sqrt{\frac{2\pi}{\sigma}}\left(1-\Phi\left(\frac{|x|}{\sqrt{2\sigma t}}
\right)\right)}\:.
$$


Finally, the case that $x=0$ can be easily verified. This then finishes the proof.


