I am attempting to compose the Maclaurin series for $\operatorname{erf}(z)$.
My disclaimer is that I am not an expert in the field of complex analysis. Below is my attempt. I am worried about convergence assumptions and / the integration term by term technique. I wonder if I am in the right direction on this?
Problem
To determine the Maclaurin series for $\operatorname{erf}(z)$.
We have, by definition
$$\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}\displaystyle \int_0^z e^{-t^2}dt$$
Since
$$e^{-z^2} = \sum_{j=0}^\infty\left(\frac{(-z^2)^j}{j!}\right),\qquad |z|<\infty $$
and (here is my worry) the convergence is uniform on the closure of $\bar{B}(\pi i , r)$ for any $r \in (0, \infty)$, I can then describe term by integration as
$$\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}\sum_{j=0}^\infty\left(\frac{(-1)^j z^{2j+1})}{j!(2j+1)}\right),\qquad |z|<\infty$$
Any pointers here if I have made mistakes would be greatly appreciated.
Best Answer
This is correct.
Both $z \mapsto e^{-z^2}$ and $z \mapsto \text{erf } (z)$ are entire functions and can be expanded in convergent power series for any $z$.
We have a general theorem that a power series can be integrated termwise along any path within the circle of convergence, which in this case has an infinite radius.