Complex Integrals - Mathematical Method

In lieu with the flow field interpretation of complex functions, generalizing the line integral for complex functions is very natural:

We can compte the integral by computing two standard real line integrals:

\begin{align*} \int_{\Gamma} f(z)dz &=\int_{\Gamma} (u+iv)(dx+idy) \\ &=\int_{\Gamma} (udx-vdy) + i\int_{\Gamma} (vdx+udy) \end{align*}

We can define $A=(u,-v)$ and $B=(v,u)$ and $r=(dx,dy)$ :

\int_{\Gamma} f(z)dz = \int_{\Gamma} A\cdot dr + i\int_{\Gamma}B\cdot dr