I don't understand how the definition of the complex logarithm was derived. It is $ log(z) = ln|z| + i Arg (z) $, where $ z = x + iy $. I've tried all sorts of method to find this definition but none of my attempts quite work out. Can someone explain this to me?
Thanks!
This is a direct extension of the real situation: if $y=e^x$ then $x=\ln y$. The exponential and natural logarithm are inverses. In the complex case we want the exponential and logarithm to be inverses as well and so we define it as such and investigate the consequences. If $z=e^w$ then $w=\ln z$ and we see that the the logarithm is defined in terms of the complex exponential's argument. If we begin with $z=x+iy=re^{it}$ you can now plug this into the definition and see why things are the way they are. The only thing remaining is to observe that Euler's formula indicated that the complex exponential is cyclic in nature and this appears in the logarithm as the different branches.