Register to reply

Cauchy Riemann Equations (basic doubt)

by ask_LXXXVI
Tags: cauchy riemann
Share this thread:
Dec20-10, 04:58 AM
P: 53
Lets say we have a function of a complex variable z , f(z).

I read that for the function to be differentiable at a point z0 , the CR equations are a necessary condition but not a sufficient condition.

Can someone give me an example where the CR equations hold but the function is not differentiable at that point , thus justifying that the CR equations holding true aren't sufficient test.

I am unable to visualise.
Phys.Org News Partner Science news on
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye
Dec20-10, 07:11 AM
Sci Advisor
PF Gold
P: 39,304
Let g(x, y)= 1 if xy is not 0, 0 if xy= 0 and let f(z)= g(x,y)(1+ i)= g(x, y)+ ig(x,y) where z= x+ iy. Then
[tex]\frac{g(x,y)}{\partial x}= \frac{\partial g(x,y)}{\partial y}= 0[/tex]
for (x,y)= (0, 0) so the Riemann-Cauchy equations are satisfied there but the function is not even continuous at (0, 0).
Dec20-10, 11:46 AM
P: 53
Thanks.Doubt resolved.

Register to reply

Related Discussions
Cauchy-Riemann equations Calculus & Beyond Homework 1
Cauchy-Riemann Equations Calculus & Beyond Homework 5
Cauchy-Riemann equations Calculus & Beyond Homework 14
Cauchy-Riemann Equations problem (f(z) = ze^-z) Calculus 3
Cauchy-Riemann Equations Calculus 0