# How to show a function is analytic?

1. Sep 6, 2010

### numberthree

how to show a function is analytic??

I know that to show a function is analytic I need to prove its differentiable, but for a fuction like log(z-i), how could i show it is analytic???

2. Sep 6, 2010

### HallsofIvy

Re: how to show a function is analytic??

Use the "Cauchy-Riemann equations which should be mentioned early in any book on "functions of a complex variable". A function f(x+ iy)= u(x,y)+ iv(x,y) is analytic at $z_0= x_0+ iy_0$ if and only if the partial derivatives, $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ are continous at the point and
$$\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$$
and
$$\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$$

3. Sep 6, 2010

### numberthree

Re: how to show a function is analytic??

yes, i know wat u mean, but i dont know how to seperate log(z-i) into u + iv form

4. Sep 6, 2010

### jackmell

Re: how to show a function is analytic??

Why not just differentiate it and then show the derivative exists in a region surrounding a point then it is analytic in that region so:

$$\frac{d}{dz} \log(z-i)=\frac{1}{z-i}$$

and that derivative exists everywhere except at z=i.

5. Sep 8, 2010

### ╔(σ_σ)╝

Re: how to show a function is analytic??

Or you could integrate the function over a closed line and show the integral is zero.

6. Mar 17, 2012

### 0911as

Re: how to show a function is analytic??

use log(z) = log(|z|) + i (arg(z))