OurBigBook.comSite Source code

Uniform L2 circle bound for a removable singularity

If

f

is holomorphic on

0 < ∣ z ∣ < R

and

sup_{0 < r < R} \int_{0}^{2 π} ∣ f (r e^{i θ}) ∣^{2} d θ < \infty,

then the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give

∣ a_{- k} ∣ = O (r^{k})

for each

k \geq 1

, so every principal-part coefficient vanishes as

r ↓ 0

.

 Ancestors (8)

 Incoming links (1)

Solution