Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$.
I am looking for a solution to this problem.
My thoughts: I know in order to find the residue of a function, I first need to find the singularities. Unfortunately this problem has me a little stumped on where to start because it didn't give me a function the way I've traditionally seen them.
If $z_0 = 0$, take the Laurent series $\displaystyle f(z) = \sum_{n=-\infty}^\infty a_n z^n$ of $f$. Then $g(z) = f(z^2) = \sum_{n=-\infty}^{\infty}a_nz^{2n}$. Hence the coefficient of $z^{-1}$ in that series of $g$, which is the residue of $g$ is $0$.