# Thread: Determine is function is Analytic

1. ## Determine is function is Analytic

I have a complex function

f(z) =ln|z| + jArg(z)

how do I show it is analytic?

2. The function...

$f(z) = \ln |z| + j \angle z$ (1)

... is the principal branch of the function $\varphi (z) = \ln z$. It is analytic in tha whole complex plane with the exception of $z=0$ where it has a branch point...

Kind regards

$\chi$ $\sigma$

3. Originally Posted by chisigma
The function...

$f(z) = \ln |z| + j \angle z$ (1)

... is the principal branch of the function $\varphi (z) = \ln z$. It is analytic in tha whole complex plane with the exception of $z=0$ where it has a branch point...

Kind regards

$\chi$ $\sigma$
Can we use Cauchy-Riemann equation to prove it is analytic except $z=0$ ?

4. Setting $z = x + j y$ and $f(z)= \ln z$ we have ...

$f(z) = u(x,y) + j\cdot v(x,y) = \frac{1}{2}\cdot \ln (x^{2} + y^{2}) + j\cdot \tan^{-1} (\frac{y}{x})$ (1)

... so that is...

$\frac{\partial u}{\partial x} = \frac{x}{x^{2} + y^{2}} = \frac{\partial v}{\partial y}$

$\frac{\partial u}{\partial y} = \frac{y}{x^{2} + y^{2}} = - \frac{\partial v}{\partial x}$ (2)

The Cauchy-Riemann condition are then satisfied in the whole complex plane with the only exception of the point $x=y=0$...

Kind regards

$\chi$ $\sigma$