# Cauchy-Riemann equations and holomorphic functions

• Apr 25th 2013, 04:25 AM
CuriosityCabinet
Cauchy-Riemann equations and holomorphic functions
Consider the function defined by $f(x+iy) = \sqrt(\left |x \right |\left |y \right |)$. Show that $f$ satisfies the Cauchy-Riemann equations at the origin yet it is not holomorphic at zero.

• Apr 25th 2013, 05:43 AM
xxp9
Re: Cauchy-Riemann equations and holomorphic functions
f(z)-f(0)/(z-0) = f(z)/z
let z goes to 0 from the positive x axis, we have f(z)/z = 0/z = 0
let z goes to 0 from the positive part of the line y=x, we have f(z)/z = x/(x+ix)=(1-i)/2
• Apr 25th 2013, 05:51 AM
CuriosityCabinet
Re: Cauchy-Riemann equations and holomorphic functions
Quote:

Originally Posted by xxp9
f(z)-f(0)/(z-0) = f(z)/z
let z goes to 0 from the positive x axis, we have f(z)/z = 0/z = 0
let z goes to 0 from the positive part of the line y=x, we have f(z)/z = x/(x+ix)=(1-i)/2

Thanks, so because the left and right-hand side limits are different, f is not holomorphic? Also, what is the first line representing?
• Apr 25th 2013, 06:44 AM
HallsofIvy
Re: Cauchy-Riemann equations and holomorphic functions
A function, of a complex variable, is "holomorphic" at a point if and only if it is differentable there. What xxp9 did is show that the limit defining the derivative does not exist by showing that the limits, as z approaches 0 from two different directions, give different resuts