I've been stuck on a particular integral I encountered. I don't need an exact solution, I doubt it even exists.
$f(x)=\frac{e^{-i (r+R-k) x} \left(i-2 e^{i (r+R) x} r x-R x+e^{2 i r x} (R x-i)\right)}{ x^3}$
I'm tasked to find$\int_{-\infty}^\infty{f^n(x) dx}$ for very large integer n and $0 < r < R$
Any suggestions on how to do so? Thanks
EDIT: ok, I've made some progress:
The following Laurent series gives f(x): $f(x)=\sum _{m=0}^{\infty } a_mx^m$ with $a_m=\frac{(i (r-R))^{2+m} R-R (-i (r+R))^{2+m}}{r (2+m)!}-\frac{i (i (r-R))^{3+m}-i (-i (r+R))^{3+m}}{r (3+m)!};$
which is related to the contour (a circle at any non-zero distance from $x = 0$) integral via $\oint_C f(x) = 2 \pi i a_{-1} =0$ when there is only one singular point.
But this was all for $n=1$, and I don't know how $\oint_C f(x)$ relates to $\int_{-\infty}^\infty{f(x) dx}$, let alone when $n\neq1$.
On $f^n(x)$, I didn't find an explicit expression for the corresponding coefficients, but did find that all coefficients are $0$ for $m < 0$. I don't know what that implies for $\int_{-\infty}^\infty{f^n(x) dx}$, please elaborate.