# Holomorphic function on the unit disc

1. May 1, 2012

### iamqsqsqs

Does there exist a holomorphic function f(z) on the unit disc and satisfies f(1/n) = f(-1/n) = 1/n^3 for every n in N?

2. May 1, 2012

### micromass

Staff Emeritus
There does not even exist a continuous function that does this.

3. May 1, 2012

### iamqsqsqs

How can we vigorously prove that? I am thinking of construct a function g such that g(1/n) = g(-1/n) = 1/n^2 and consider f/g to do it. However I am stuck and cannot go on

4. May 1, 2012

### micromass

Staff Emeritus
What would f(0) be?

5. May 1, 2012

### Citan Uzuki

Last time I checked, $z \mapsto |z|$ was continuous...

iamqsqsqs, try looking at the zeros of f(z) - z^3. Do they form an isolated set of points?

6. May 1, 2012

....

7. May 1, 2012

### Citan Uzuki

Sorry, brain fart. I meant to say $z \mapsto |z|^3$

8. May 1, 2012

### DonAntonio

Hehe...yes, I supposed so. Happens to me all the time. Your answer to look at the zeroes of $\,\,f(z)-z^3\,\,$ pretty much wraps this up, though.

DonAntonio