I have an integral that can be solved via recursive integration by parts. In my case, $\mathrm{d}v=e^{-ax^{2}}$. Question: Is there a solution or special function defined as the n-th indefinite integration of the Gaussian function?
\begin{equation} \mathbf{i}^{n} \exp\{-ax^{2}\}= \mathop{\int\dots\int}\limits_{n \ \text{times}} e^{-ax^{2}} \ \mathrm{d}x^{n} = \ ? \end{equation}
Anyways, I am aware that there is the solution for this problem http://dlmf.nist.gov/7.18 for the complementary error function:
\begin{equation} \mathbf{i}^{n}\ \mathrm{erfc}\, z= \int_{z}^{\infty} \mathbf{i}^{n-1} \mathrm{erfc}\, t \ \mathrm{d}t = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty} \frac{(t-z)^{n}}{n!} e^{-t^{2}} \ \mathrm{d}t \end{equation}
I thought this may be useful. Any help would be great.