You would need to check the Cauchy-Riemann equations.
Suppose is holomorphic on the open disc of the complex plane.
How would one go about proving is holomorphic in ?
I attepted this by writing out the definition of the derivative, substituting doing the algebra and using the facts that the real and imaginery parts of are differentiable and if then
Is this the way this question would be tackled?? i'm not sure if my proof is a hundred percent correct.
Also I want to prove is differentiable at iff f'(a) = 0.
I'm a bit stuck on this part. Any ideas?
Incidently this is question 5.11 from Introduction to Complex Analysis by Priestly.
how can i check the cauchy conditions? i have no information about the continuity of the partial derivatives of the real and imaginary parts of f. I only know that f is holomorphic on the open disc and thus continuous but the cauchy equations only provide a sufficient condition for differentiability when the partial derivatives are continuous.
In general if is an open set and is holomorphic then is holomorphic in , and to see this just notice .
For the second one use the Cauchy-Riemann equations together with the definition of the conjugation operation to conclude that the partials must be zero.
Can you descirbe how the limit operation is performed please. I dont see how to take the limit of the lhs. Is this correct:
I've used the fact is a continuous function so I can bring the limit inside the bracket. Is that right?