# How to show a function is analytic?

## Main Question or Discussion Point

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???

HallsofIvy
Homework Helper

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
$$\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$$

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

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???
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.

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

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