Consider the definite integral $$ F(n,y,z) := \int_{y}^{\infty} \frac{2x^{n+1}}{x^2-z^2} dx \ . $$ for parameters $y>z>0$ and $n \in \mathbb{C}$.
I want to calculate $F(n,y,z)$ as a series about $n=0$, where notice that $F$ is divergent at the point $n=0$(the integral only converges for $\mathrm{Re}[n]<0$) . Mathematica tells me $$ F(n,y,z)= z^n \ \mathrm{B}_{\tfrac{z^2}{y^2}}\left( - \frac{n}{2}, 0 \right) \ $$ where $B_{x}(a,b) := \int_0^x u^{a-1} (1-u)^{b-1} du$ is the incomplete Beta function.
I am unable to expand this function about $n=0$ using this form, although I have been able to figure out that it scales as $1/n$. Can someone help with this?
EDIT: Apologies, I had a typo and forgot the limits on this definite integral.
It may be helpful to expand this function as a series of both $\log x$ and $x$. $$\int \exp(n\log x) \left( \frac{1}{x+z}+\frac{1}{x-z} \right) dx=\int \sum_{k=0}^\infty \frac{n^k(\log x)^k}{k!} \cdot \frac{2x}{x^2-z^2} dx$$ and prove the integral and summation exchanges.