Let $f,\phi$ be two functions holomorphic on the open simply connected $\Omega$ with $0\in\Omega, \phi$ not identicaly $0$ with $f(z)=\phi(z)h(z)$, $h$ holomorphic in $\Omega$ and $g(z)=\frac{1}{z}\int_0^z\frac{f(j)}{\phi(j)}dj$.
1) Prove that $g$ is holomorphic in $\Omega$ and
2) Calculate the coefficients of the Taylor expansion of $g$ as a function of the coefficients of the Taylor expansion of $f$ and the Laurent coefficients of $\phi$ around $0$.
I proved that $g$ is holomorphic by Morera's Theorem. Let $g(z)=\sum_{n=0}^{\infty}b_nz^n, f(z)=\sum_{n=0}^{\infty}a_nz^n$ and $1/\phi(z)=\sum_{n=-N}^{\infty}\gamma_nz^n$ then I have $(zg(z))'=\frac{f(z)}{g(z)} \implies \sum_{n=0}^{\infty}(n+1)b_nz^n=(\sum_{n=0}^{\infty}a_nz_n)(\sum_{n=-N}^{\infty}\gamma_nz^n)$. Taking the positive and negative exponents separately: $\sum_{n=0}^{\infty}(n+1)b_nz^n=\sum_{n=0}^{\infty}(\sum_{k=0}^na_k\gamma_{n-k})z^n+(\sum_{n=0}^{\infty}a_nz_n)(\sum_{n=-N}^{-1}\gamma_nz^n)$.
So my question is: how can I find the coefficients of this sum: $\sum_{n=0}^{\infty}(\sum_{k=0}^na_k\gamma_{n-k})z^n+(\sum_{n=0}^{\infty}a_nz_n)(\sum_{n=-N}^{-1}\gamma_nz^n)$ ?