How do I integrate this in terms of error function

Question

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think I am missing something.

Answer 1

The change of variable $z=\dfrac{\xi-y}{2\sqrt t}$ transforms the integral into $$ \int_{-\infty}^{\xi/(2\sqrt t)}(\xi-2\,\sqrt t\,z)e^{-z^2}\,dz=\xi\int_{-\infty}^{\xi/(2\sqrt t)}e^{-z^2}\,dz-2\,\sqrt t\int_{-\infty}^{\xi/(2\sqrt t)}z\,e^{-z^2}\,dz. $$ The first integral can be expressed in terms of $\text{erf}(\xi/(2\sqrt t)$, and the second can be computed explicitly.