Research proof index

The Residue Theorem on a rectangle for functions with simple poles.

A vertical splitting identity for rectangular contour integrals. Under integrability of $f$ along the two horizontal sides, splitting the rectangle with real coordinates from $x_0$ to $x_1$ at $a$ gives $\int_{\partial R(x_0,x_1;y_0,y_1)}f=\int_{\partial R(x_0,a;y_0,y_1)}f+\int_{\partial R(a,x_1;y_0,y_1)}f.$ The two integrals over the shared vertical side cancel.