Can a Non-Constant Holomorphic Function Equal Zero Everywhere?

[OhNoYaDidn't]

Homework Statement

With

. Give an example, if it exists, of a non constant holomorphic function

that is zero everywhere and has the form 1/n, where n € N.

Homework Equations

So.. This was in my Complex Analysis exam, and i have no idea what to do. I always seem to get stuck at these more abstract questions.

 

[FactChecker]

It seems like your problem statement must be wrong. How can a function be zero everywhere and nonzero at some points? Am I reading it wrong?

 

[WWGD]

Just declare the function to be $0$ on a subset of the plane that has a limit point. But yes, the statement is kind of confusing.

 

[OhNoYaDidn't]

Homework Statement

With

. Give an example, if it exists, of a non constant holomorphic function

that returns zero in all points, and has the form 1/n, where n € N.

Homework Equations

So.. This was in my Complex Analysis exam, and i have no idea what to do. I always seem to get stuck at these more abstract questions.

I tried to fix it.
I'm so sorry, but i translated this. But even in my mother togue it's confusing.

 

[WWGD]

The only way I can make sense of it is this: find an analytic function that has zeros at all points $1/n, n=1,2,$ and so that $f(0)=0$. And then you use the identity theorem to conclude f must be identically $0.$

 

[FactChecker]

I bet they want a function that is zero at the points 1/n, analytic on the right half plane, and not constant.

 

[RUber]

This must be some kind of a trick, right? So satisfy the conditions one by one, and then satisfy the "every point" condition by making it the limit of some sequence.
1) Holomorphic - start thinking sine / cosine
2) looks like 1/n at x = n, so maybe $\frac{1}{x}$ times some sine or cosine function
3) returns zero at every point, maybe the limit of your frequency component of the sine or cosine, so that at every point, you will have a vertical line from -1/x to 1/x.