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(z−z0)n has radius of convergence R>0. Then the function f:BR(z0)→C defined by
f(z):=∞∑n=0an(z−z0)n
is differentiable in BR(z0) and for any z∈BR(z0) the derivative is given by the formula
f′(z)=∞∑n=1nan(z−z0)n−1.
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
No comments:
Post a Comment