In my research, I am trying to evaluate an integral in closed form. I realized that if I can write the following function:
$$ \text{erf}\left(x\right)^{k+1}$$
as a Taylor expansion centered at zero, then I can evaluate the integral in series form, which would be great! I used Mathematica to find the derivatives and evaluated them at x=0 and got the following:
\begin{equation} \[\begin{array}{l} k = 0\\ \frac{{{d^n}}}{{d{x^n}}}{\left[ {{\rm{erf}}{{\left( x \right)}^{k + 1}}} \right]_{x = 0}} = \left\{ {0,\frac{2}{{\sqrt \pi }},0, - \frac{4}{{\sqrt \pi }},0,\frac{{24}}{{\sqrt \pi }},0, - \frac{{240}}{{\sqrt \pi }},0,\frac{{3360}}{{\sqrt \pi }},0, \ldots } \right\}\\ k = 1\\ \frac{{{d^n}}}{{d{x^n}}}{\left[ {{\rm{erf}}{{\left( x \right)}^{k + 1}}} \right]_{x = 0}} = \left\{ {0,0,\frac{8}{\pi },0, - \frac{{64}}{\pi },0,\frac{{896}}{\pi },0, - \frac{{18432}}{\pi },0,\frac{{509952}}{\pi },0,-\frac{17940480}{\sqrt{\pi}},0,\frac{771194880}{\sqrt{\pi}},0,-\frac{39305871360}{\sqrt{\pi}} \ldots } \right\}\\ k = 2\\ \frac{{{d^n}}}{{d{x^n}}}{\left[ {{\rm{erf}}{{\left( x \right)}^{k + 1}}} \right]_{x = 0}} = \left\{ {0,0,0,\frac{{48}}{{{\pi ^{3/2}}}},0, - \frac{{960}}{{{\pi ^{3/2}}}},0,\frac{{25536}}{{{\pi ^{3/2}}}},0,\frac{{ - 895488}}{{{\pi ^{3/2}}}},0, \ldots } \right\}\\ k = 3\\ \frac{{{d^n}}}{{d{x^n}}}{\left[ {{\rm{erf}}{{\left( x \right)}^{k + 1}}} \right]_{x = 0}} = \left\{ {0,0,0,0,\frac{{384}}{{{\pi ^2}}},0, - \frac{{15360}}{{{\pi ^2}}},0,\frac{{688128}}{{{\pi ^2}}},0,\frac{{ - 37355520}}{{{\pi ^2}}}, \ldots } \right\} \end{array}\] \end{equation}
The goal is to find the Taylor Expansion of each k, and then hopefully find a general formula which relates them all. I will address convergence after this is done.
For k = 0, if we start a new index $n$ at $n=1$,the pattern seems to be recursive: $$a_1 = 2$$ $$a_n = 2(2n-3)a_{n-1}\;\;\;\;\;\;n\ge 2$$
Thus the taylor expansion is: $$ \text{erf}\left(x\right) = \frac{1}{\sqrt{\pi}}\sum_{n=1}^{\infty} \frac{(-1)^{n+1}a_n}{(2n-1)!}x^{2n-1}$$
I plotted this against the function and verified that it does indeed seem to be correct. I am stuck at k=1. Can anyone help identify any of the above patterns for k>0?
For k=1, at first glance it seems very similar, but breaks down by the 4th term. Namely (previous term in parenthesis), first term is 8, then next is 8*(8), next is 14*(8*8) so seems like multiplier is going up by 6 each time. But the next term is 17920 which is wrong.
Any thoughts?