Let $f$ be an entire function on the complex plane.

Is the radius of convergence of $f$ around any point $z_0$ infinite? If so, why?

Thank you.

  • $\begingroup$ What are your thoughts on this question? $\endgroup$ – abiessu Jun 22 '14 at 18:43

A function $f: \mathbb{C} \to \mathbb{C} $is said to be entire if it is holomorfic in the all complex plane, but using the Cauchy's Integral Formula, we have that a holomorfic function in a region is also analytic in this same region region. Therefore, therefore an entire function $f$ is analytic in the all complex plane, but we also have that the radius of convergente of a series centered in a point $z_0$ is the distance between $z_0$ and the nearest singularity of the series, in that case, it is going to be infinite.


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