Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 8E (Further Complex Methods)

The numbers $z_j$ are the nonzero zeros of a function $f$, repeated according to their multiplicity. The displayed Weierstrass factorization theorem formula uses genus-one Weierstrass elementary factors and a linear exponential factor; it applies under the corresponding growth assumptions, rather than to every arbitrary entire function.

Apply it to the entire function $g(z)=\sin (\pi z)/(\pi z)$, with its removable value $g(0)=1$. Its zeros are $\pm 1,\pm 2,\dots$, each simple, and $g^{\prime }(0)=0$ because $g$ is even. Pairing the factors at $n$ and $-n$ givessoThe product converges uniformly on compact subsets because $\sum n^{-2}<\infty$. Away from the integers its logarithmic derivative can be taken term by term: the resulting series also converges locally uniformly, its terms being $O(n^{-2})$. HenceThis is an identity of meromorphic functions, understood away from their poles.