# Nice Complex Analysis Problem

• Oct 11th 2007, 07:31 PM
ThePerfectHacker
Nice Complex Analysis Problem
As I said I was taking an exam more than a month ago on Complex Analyis. One of the optional problems was (if I remember correcty):

"Find all complex rational functions which have modulos 1 on the unit circle"*.

Note: I cannot help you with this problem it uses a theorem which I never learned.

*)I was told that this problem was a favorite problem to give as an examination problem to graduate students in a certain event (which I forgot the name of).
• Oct 12th 2007, 12:06 AM
Opalg
It sounds as if they are asking you to rediscover the theory of Blaschke products.
• Oct 12th 2007, 10:13 AM
ThePerfectHacker
Quote:

Originally Posted by Opalg
It sounds as if they are asking you to rediscover the theory of Blaschke products.

This is exactly the term a professor told me it is. Can you post a proof?

And do you think this is a too hard of a question for an exam?
• Oct 13th 2007, 01:11 AM
Opalg
Here's a bare outline of how the proof goes. The set G of all complex rational functions which have modulus 1 on the unit circle is a group (under pointwise multiplication). For each complex number a with |a|<1, the function $\displaystyle f_a(z) = \frac{z-a}{1-\bar{a}z}$ is in G.

Now suppose that f is in G. If f has no zeros in the unit ball then f must be constant. (That is the tricky part of the argument. According to the Wikipedia page that I linked to in my previous comment above, "this fact is a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|f(z)|)".)

If f does have zeros on the unit ball, call them a_1, a_2, ..., a_n (counted according to their multiplicity). Then $\displaystyle f/(f_{a_1}f_{a_2}\cdots f_{a_n})$ is a function in G with no zeros in the unit ball and is therefore constant. Hence $\displaystyle f = \lambda f_{a_1}f_{a_2}\cdots f_{a_n}$ for some number $\displaystyle \lambda$ with $\displaystyle |\lambda|=1$.

That's the structure theorem for functions in G. Whether or not it's fair to ask it as an exam question depends on the students' background, I guess. I would say it's a pretty tough assignment for any student who hasn't previously seen the result.
• Oct 13th 2007, 04:45 PM
ThePerfectHacker
The quick outline of the solution to this problem was presented to me using Schwartz-Reflection Principle.