Its late at night and I've somehow convinced myself that every holomorphic function must be zero. Please tell me why I'm being an idiot.
Cauchy's integral theorem says that the integral of a holomorphic function $f$ around a closed curve equals zero.
Cauchy's integral formula says for any circle $C$, any $a$ enclosed by the circle satisfies $f(a)=\frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a}dz$. The integrand is holomorphic so the integral is zero.