We can see that a holomorphic functions are the complex functions such that they can express a convergent power series
We want to compute . Suppose is closed curve (Jordan curve).
We compute the integral for . Firstly, consider . Change the variables, , so
This computation, we think that it is important, because it implies a classical result
Theorem 1 (Cauchy integral theorem) Let be a holomorphic function on . Then
with is closed Jordan curve.
Of course it has modulo an any closed Jordan curve: If and are two closed Jordan curve with a common fixed point then