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# Complex Differentiable Function watch

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1. Well I need to find when a complex function is complex differentiable however this question is a little more complicated.

I'm just stuck as surely there are infinitely many solutions? Or have I made a silly mistake...
2. (Original post by TomLeigh)
Well I need to find when a complex function is complex differentiable however this question is a little more complicated.

I'm just stuck as surely there are infinitely many solutions? Or have I made a silly mistake...
3. (Original post by ztibor)
-.- I remember that from the lectures now, just read it wrong :P
So this surely gives us y = 3x^2 - 2 so what are the actual points in the complex field in which the function is complex differentiable?
4. The complex numbers z = x+iy where y = 3x^2 - 2 are the only points that are complex differentiable.
5. Thanks, I wasn't sure if the answer was as simple as that (just explaining) or if there were more steps required. I'll pos rep you tomorrow
6. (Original post by Creole)
The complex numbers z = x+iy where y = 3x^2 - 2 are the only points that are complex differentiable.
The question then asks for which points in complex field is the function analytic. What does this mean exactly? Most examples use a simple sentence but I'm not sure what they mean eg:

Since the imaginary axis contains no non-empty open disc, f is analytic nowhere

for another question.
7. I believe it is nowhere analytic, although I found the notes quite difficult to understand (at this point I will reveal that I am doing the same module as you ).

The key being that there is no open disc upon which the function is differentiable in the complex plane.
8. (Original post by TomLeigh)
-.- I remember that from the lectures now, just read it wrong :P
So this surely gives us y = 3x^2 - 2 so what are the actual points in the complex field in which the function is complex differentiable?

The Cauchy Riemann equations are a nesacery not sufficient condition for differentiabilty at a point. You can't use them to prove a function is differentiable at at point, only that it is not differentaible at a point if the CR euqations don't hold.

There is a partial converse which says that
Let be open and . Then if u and v have continuous first order partial derivitives in G and that they exist satisfy the CR equations at z, then f ii differentiable at z.

a function is analytic in an open set G if it is differentiable at all points in G

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