Good afternoon. I have the following integral that I need help integrating;
$$F(x)=\int_0^\infty e^{s\left(j-\frac{1}{x}\right)}\left[T_N(s)\right]^{k-j}ds$$
where $T_N(s)$ is the truncated exponential function $$T_N(s)=\sum_{n=0}^N\frac{s^n}{n!}$$
and $j,k$ are whole numbers
I figured that tackling this using integration by parts is the best choice, but naturally this could take a while given that my choice of $u_1=T_N(s)^{k-j}$ yields a $du_1$ of $$du_1=(k-j)T_N(s)^{k-j-1}T_{N-1}(s)ds$$ since $\frac{d}{ds}T_N(s)=T_{N-1}(s)$. This does not seem to be simplifying the problem though as the next iteration would require me to then choose my $u_2$ to be $$u_2=\frac{1}{(k-j)}\frac{du_1}{ds}=T_N(s)^{k-j-1}T_{N-1}(s)$$
Are there any other approaches to this without this drawn out IBP method or without considering expanding the truncated exponential?
from Hot Weekly Questions - Mathematics Stack Exchange
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