Holomorphic function on the unit disc

[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?

 

[micromass]

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

 

[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

 

[micromass]

What would f(0) be?

 

[Citan Uzuki]

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

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?

 

[DonAntonio]

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


*** Last time I checked $\,\,\displaystyle{\left|\frac{1}{n}\right|\neq \frac{1}{n^3}}$ ...

DonAntonio ***


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

...

 

[Citan Uzuki]

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

 

[DonAntonio]

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

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