Let $A(z) = \sum_{n\ge0} a_n z^n$ and $B(z) = \sum_{n\ge0} b_n z^n$ be two meromorphic functions analytic at the origin. Let $S_A$ and $S_B$ denote the set of singularies of the functions $A$ and $B$ respectively. We define $$A\star B(z) = \sum_{n\ge0} a_n b_n z^n.$$ The Hadamard multiplication theorem tells us that $$ A\star B(z) = \frac{1}{2 \pi i} \int_{c} \frac{A(\zeta)}{\zeta} B(z/\zeta) d\zeta $$ for $|z|$ small enough and that this integral can be continued to the star domain that we get by removing the points that correspond to the products of singularities of $A$ with singularities of $B$, i.e. the points $S_{A\star B}=\{z \in \mathbb{C}:z=z_1\cdot z_2,\, z_1\in S_A,\, z_2\in S_B \}$.
In a recent discussion I had I was told that this integral can be continued even further and on all Riemann sheets set of possible singularities is the set $S_{A\star B}\cup\{0,\infty\}$. However the person that told me that could not locate a reference for this.
Is this true? Is there a reference for this? Can we say something when $A$ and $A$ are allowed to have branching singularities?
The keyword is substract the poles. Assume for simplicity they are of order $1$
Also note that $$F\star \frac{1}{1-z/c}= \sum_{n=0}^\infty f_n c^{-n} z^n = F(z/c)$$
For some $R > 0$, let $\rho$ be the poles of $A(z)$ and $A_R(z)$ analytic on $|z| < R$ such that $$A(z) = A_R(z) + \sum_{|\rho| < R} \frac{C_\rho}{1-z/\rho}$$
The same for $$B(z) =B_R(z) + \sum_{|\tau| < R} \frac{D_\tau}{1-z/\tau}$$
Hence $$A \star B(z) = A_R \star B_R(z) + \sum_{|\tau| < R, |\rho| < R}\frac{C_\rho D_\tau}{1-z/(\rho\ \tau)} + \sum_{|\tau| < R}D_\tau A_R(z/\tau)+ \sum_{|\rho| < R}C_\rho B_R(z/\rho)$$
Since $A_R,B_R$ are analytic on $|z| < R$ then $A_R \star B_R$ is analytic on $|z| < R^2$ so that $A \star B(z)$ is meromorphic with poles of order $1$ at $ \rho\ \tau$