I am learning a bit of complex function theory on my own. I found a theorem in a paper that I can't seem to find a proof of. It reads as follows:
A complex function $f$ is holomorphic if and only if $\frac{\partial f}{\partial \overline{z}}=0$.
I understand that the differential operator is defined as $\frac{\partial}{\partial \overline{z}}=\frac{1}{2}\left(\frac{\partial}{\partial u}+i\frac{\partial}{\partial v}\right)$. But how does one actually prove the above theorem? Thanks in advance for any help!
This follows directly from the Cauchy-Riemann equations. By demanding that this differential operator is zero, you are simply demanding that the Cauchy-Riemann equations hold.