I am given the function $f:\mathbb{C}\setminus \{0, -1, -2, \ldots \}\to\mathbb{C}$ that is holomorphic on its domain, such that $f(1)=1$ and $f(z+1)=zf(z)$. I want to show that $0, -1, -2, \ldots$ are simple poles of $f$ and that the residues are given by $(-1)^n/n!$ at $z=-n$. Any hints? I am thinking of using a Laurent series around each integer and show that there is only one reciprocal term, but I don't know how to incorporate the given that $f(1)=1$. Any hints?
2026-03-28 02:48:53.1774666133
Show that $f$ has a simple pole at $0, -1, -2, \ldots$.
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