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Cauchy Riemann Equations (basic doubt)

by ask_LXXXVI
Tags: cauchy riemann
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ask_LXXXVI
#1
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.
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HallsofIvy
#2
Dec20-10, 07:11 AM
Math
Emeritus
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Thanks
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).
ask_LXXXVI
#3
Dec20-10, 11:46 AM
P: 53
Thanks.Doubt resolved.


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