A function f: G $\rightarrow$ $\Bbb C$ has period p $\in$ $\Bbb C$ if f(z+p) = f(z) for all z $\in$ G.
Let G = $\Bbb C$ $\setminus$ {m + ni | m, n $\in$ $\Bbb Z$} and suppose f: G $\rightarrow$ $\Bbb C$ is a holomorphic function with periods 1 and i.
Show that if the origin is neither an essential singularity, nor a pole of order 2 or higher, then f must be constant.
I understand an essential singularity means that the power series has negative coefficients, so we are looking for an origin that is nonnegative and there will only be one singularity?