Complex integral is zero I am doing some complex analysis and I am working on a question, it seems blantanly easy for the answer which is what makes me sceptical. So I would like a second input Let $C$ be the rectangle with verticies at $\pm b$ and $\pm b+ai$ with $a,b>0$. I am to explain why this holds true $$\int_C e^{-z^2}dz=0$$ In the papers they use a simplified version of Cauchy's integral theorem but to me that would be the reason. Is that really it or am I missing something? If I am missing something please do enlighten me

It's a direct application of the integral theorem. $e^{-z^2}$ is analytic everywhere.