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Complex Analysis

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humph:

--- Quote from: /0 on July 29, 2010, 10:04:22 pm ---Thanks zzdfa xD


When people talk of a function being differentiable, do they mean differentiable in a neighbourhood, or differentiable everywhere?
Usually people only test differentiability at (0,0), is there any reason why?

The lecture notes give examples or functions which are non-differentiable:
, , , and
Does this mean differentiable nowhere, or does it mean not differentiable at a selection of points?


Also, the tute says that is not analytic at . But

So , , , .

So at , the Cauchy Riemann equations are satisfied, so why isn't the function analytic at z = 0?

Thanks

--- End quote ---
Who's your tutor? As was mentioned, it's differentiable at zero, but analytic (well I prefer holomorphic) means it has to be differentiable in a neighbourhood of zero, which isn't the case here.

/0:
My tutor's Davidson Ng, but we haven't had a tutorial yet.

I understand what you mean, that it must be differentiable in a neighbourhood... but the theorem that I used is as follows:

Let be given by , for . Assume that all partial derivatives , , , exist, are continuous, and satisfy the Cauchy-Riemann equations at . Then is holomorphic at .


It never says anything about being differentiable in a neighbourhood, only about the continuity of the partial derivatives

Ahmad:
Where did you see that?

satisfies CR equations at z = 0 and nowhere else, the partials are continuous but f is nowhere analytic.

/0:
I believe it's called the Looman-Menchoff Theorem

The version I quoted is what's in the lecture notes. Is there a difference?

Ahmad:
Yeah. The result you should remember is that if CR equations are satisfied at a point and the partials are continuous at the point then the function is differentiable at the point. Then being holomorphic/analytic in some open set just means differentiable everywhere in the open set, so CR equations are satisfied + continuous partials everywhere in the open set. Analytic at a single point would mean differentiable in some open set containing the point.

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