I have a complex function
f(z) =ln|z| + jArg(z)
how do I show it is analytic?
The function...
$\displaystyle f(z) = \ln |z| + j \angle z $ (1)
... is the principal branch of the function $\displaystyle \varphi (z) = \ln z$. It is analytic in tha whole complex plane with the exception of $\displaystyle z=0$ where it has a branch point...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Setting $\displaystyle z = x + j y$ and $\displaystyle f(z)= \ln z$ we have ...
$\displaystyle f(z) = u(x,y) + j\cdot v(x,y) = \frac{1}{2}\cdot \ln (x^{2} + y^{2}) + j\cdot \tan^{-1} (\frac{y}{x}) $ (1)
... so that is...
$\displaystyle \frac{\partial u}{\partial x} = \frac{x}{x^{2} + y^{2}} = \frac{\partial v}{\partial y}$
$\displaystyle \frac{\partial u}{\partial y} = \frac{y}{x^{2} + y^{2}} = - \frac{\partial v}{\partial x}$ (2)
The Cauchy-Riemann condition are then satisfied in the whole complex plane with the only exception of the point $\displaystyle x=y=0$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$