Friday, 28 October 2016

derivatives - How to prove that a complex power series is differentiable




I am always using the following result but I do not know why it is true. So: How to prove the following statement:



Suppose the complex power series n=0an(zz0)n has radius of convergence R>0. Then the function f:BR(z0)C defined by
f(z):=n=0an(zz0)n


is differentiable in BR(z0) and for any zBR(z0) the derivative is given by the formula
f(z)=n=1nan(zz0)n1.




Thanks in advance for explanations.


Answer



It is a result of Abel that says that the power series converges in the ball BR(z0). Moreover since there is uniform convergence in every concentric sub-ball, Br(z0) with $r

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