# On a property of holomorphic functions

We can see that a holomorphic functions are the complex functions such that they can express a convergent power series

$\displaystyle f(z) = \sum_{i=0}^\infty a_nz^n.$

We want to compute ${\int\limits_\gamma f(z)dz}$. Suppose ${\gamma}$ is closed curve (Jordan curve).

We compute the integral for ${\gamma = \mathbb{S}^1}$. Firstly, consider ${\int_{\mathbb{S}^1}z^ndz, n \ge 1}$. Change the variables, ${z = e^{2\pi it}, t \in [0,1]}$, so

$\displaystyle \int_{\mathbb{S}^1}z^ndz = \int_0^1e^{2n\pi it}2 \pi i\cdot e^{2\pi it}dt = \int_0^1e^{2(n+1)\pi it}2 \pi idt = \dfrac{e^{2(n+1)\pi it}}{n+1}\bigg|_0^1 = 0.$

This computation, we think that it is important, because it implies a classical result
Theorem 1 (Cauchy integral theorem) Let ${f : \mathbb{C} \rightarrow \mathbb{C}}$ be a holomorphic function on ${\mathbb{C}}$. Then

$\displaystyle \int_\gamma f(z) dz= 0,$

with ${\gamma}$ is closed Jordan curve.

Of course it has modulo an any closed Jordan curve: If $\gamma_1$ and $\gamma_2$ are two closed Jordan curve with a common fixed point then  $\displaystyle \int_{\gamma_1} f(z)dz = \int_{\gamma_2} f(z) dz,$