I am studying complex analysis. In the beginning of the chapter on integration theory there is a theorem known as Cauchy's theorem. It states that:
If $\Omega\subset \mathbb C$ be open and $f:\Omega\to \mathbb C$ be a holomorphic function and $C$ be a closed curve in $\Omega$ whose interior is also in the set, then $\int\limits_C f(z)dz=0$.
I want to know the significance of this theorem. I have tried to understand on my own but I want to make sure if it is all I need to understand. Cauchy's theorem basically says that if I integrate a holomorphic function along a closed curve, then the integral will be $0$, this basically means that given any two points in the set $\Omega$, I can choose whichever path I want between them for integrating between the two points without affecting the result. That means integral of a holomorphic function is path independent. Am I correct? Is there anything I am yet to realize about this theorem?
Best Answer
What you did is mostly correct. There is however one important thing missing: yes, if you have two points $z_0$ and $w_0$ and if you take two paths $\gamma$ and $\mu$ from $z_0$ to $w_0$, then the integrals $\int_\gamma f(z)\,\mathrm dz$ and $\int_\mu f(z)\,\mathrm dz$ will be equal provided that you can find an homotopy $H$ between them such that the range of $H$ is contained in $\Omega$. To be more precise: here, an homotopy is a continuous map from $[a,b]\times[0,1]$ (where $[a,b]$ is the domain of $\gamma$ and $\eta$) such that
That is, you can deform (in a continuous way) the path $\gamma$ into the path $\eta$ without leaving $\Omega$ and in such a way that all the intermediate paths (those of the form $t\mapsto H(t,u)$, for a fixed $u\in[0,1]$) go from $z_0$ to $w_0$. Then, yes, the integral will be the same for both paths.