I am asked to evaluate this integral using residues.
$$\int_0^1 \frac{1}{x^\omega(1-x)^\omega} \, dx$$ where $$0<\omega<1$$
I'm thinking since I have branch points at $0$ and $1$ that I ought to integrate over a contour that goes around a branch cut $[0,1]$. However, I'm confused as to how to decide on a function and the values of the arguments above and below the cut. Any help is appreciated!
(Disclaimer: I am not using complex analysis here.)
Note that the Beta function is defined as $$\operatorname B(z_1+1,z_2+1)=\int_0^1 x^{z_1}\cdot(1-x)^{z_2} \,\mathrm dx$$ for two complex numbers with real part $>-1$.
Hence, your integral is $$\int_0^1 x^{-\omega}\cdot(1-x)^{-\omega}\,\mathrm dx=\operatorname B(1-\omega,1-\omega)=\frac{\Gamma(1-\omega)^2}{\Gamma(2-2\omega)},$$ where $\Gamma$ denotes the Gamma function.
My derivation here used complex analysis at most implicitly though (by using results on the Beta/Gamma function). I don't see any way to simplify the last expression, so I am not sure if you can reach anything nice using contour integration.