If a function $E(z)$ is defined on $\mathbb{C}$ by $$ E(z) = \int_0^z e^{-w^2} dw,$$ find a power series expansion for $E(z)$ about $0$. What does this power series converge?
I know how this should go. I need to find a power series expansion of $e^{-w^2}$, and then integrate term by term to find the final result.
In class, we usually used some kind of trick to transform the integrand into a geometric series, but I can't quite see how to do that here. Any hints or suggestions as to this first step would be most appreciated.
Do you know the power series for $e^x$? Try plugging $-w^2$ into it...