- #1

Nusc

- 760

- 2

## Homework Statement

If G = { z in C: 0<|Z|<1} show that every f in L_{a}^{2}(G) has a removable singularity at z = 0

Proof:

We must show that lim z->0 z*f(z) = 0 for all f in L_{a}^{2}(G)

By a corollary 1.12, if f in L_{a}^{2}(G), a in G and 0<r<dist(a,bdr G), thne

|f(a)| <= 1/(r\sqrt(\pi))||f||_2,

for |z| < 1/2 we have that

|f(z)| <= 1/(|z|\sqrt(pi)/sqrt(2)) ||f||_2 = 2/(|Z|sqrt(pi) ||f||_2

so that |zf(z)| = |z||f(z)| <= 2/sqrt(pi) ||f||_2.

how do I proceed from there