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This question was part of my assignment which couldn't be discussed due to pendamic.

If f $\in H(0<|z|<R)$ and $\int_{0<x^2+y^2<R} {|f(x+iy)|}^2 dx dy <\infty$ prove that f has either a removable singularity or a pole of order 1 at 0.

I have studied my class notes thoroughly but I am clueless on which result should I use in this problem.

function f is Holomorphic on disc 0<|z|<R . Removable singularity is a singular point of a function for which it is possible to assign a complex number in such a way that becomes analytic while pole of order 1 means that z f(z) is analytic as z approaches 0.

Can anyone please tell how should I approach this question ?

Thanks!!

  • Does this answer your question? https://math.stackexchange.com/questions/63458/singularities-in-the-punctured-unit-disc-and-square-integrability – Sumanta Sep 17 '20 at 15:18
  • Also, https://math.stackexchange.com/questions/3290604/a-singularity-at-0-is-removable-if-the-complex-function-is-square-integrable – Sumanta Sep 17 '20 at 15:19

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