Try completing the square and using the error function. It's a really messy integral.

$\newcommand{\erf}{\operatorname{erf}}$

Yes, you need to consider $\erf(z)$ as an elementary function (and there is no compelling arguments why trigonometric functions are considered as such but not many others. In fact there are plenty of important special functions coming often from PDE, more precisely, from separation of variables--not $\erf$ but many others).

When you integrate erf(z) it always gives you zero because it's an odd function. When multiplied to any integrated function (and as alpha -> 0), the resulting functions are always 0. Does that make any sense?

The error function itself isn't integrated here, the function G(x,y,t)*g(y) is integrated.

It's true that an odd function will integrate to 0 on a domain symmetric about 0, and anything multiplied by this evaluated integral will be 0, but because we're multiplying G(x,y,t) inside of the integral sign that fact isn't too relevant here.

Consider the analogue to the function f(x) = x. f is an odd function, so the integral on say (-a,a) is 0. But when we multiply inside of the integral sign by another odd function g(x) = x^3, we have g*f(x) = x^4- certainly not 0 on the interval (-a,a) when a != 0.