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Fall 2011 - Problem 9

mean value property

Let D={z∈C∣∣z∣<1}\D = \set{z \in \C \mid \abs{z} < 1}, and suppose that ff is a holomorphic function in the punctured open unit disk D∗=D∖{0}\D^* = \D \setminus \set{0} such that

∫D∗∣f(z)∣2 dλ(z)<∞,\int_{\D^*} \abs{f\p{z}}^2 \,\diff\lambda\p{z} < \infty,

where integration is with respect to 22-dimensional Lebesgue measure λ\lambda. Show that ff has a holomorphic extension to the unit disk D\D.
Solution.

See Fall 2011 - Problem 9 and apply Riemann's removable singularity theorem.