Problem. Suppose that $f$ has an isolated singularity at the point $a$, and $f'/f$ has a first-order pole at $a$. Prove that $f$ has either a pole or a zero $a$
This is a problem from a past qualifying exam in the institution I'm attending now which might appear in the final exam on tomorrow.(so if asking this is a problematic, then let me know, I'll delete it.)
Because $f$ should have pole or removable or essential singularity at $a$, I checked that if $f$ has removable singularity at $a$ then $f(a)$ has to be $0$ to satisfy the condition in the problem, and also checked that if $f$ has pole at $a$ then $f$ satisfies the condition in the problem.
So, I was trying to show that $f$ can't have essential singularity at $a$, and tried to use Casorati-Weiestrass theorem to derive some contradiction and it didn't work. I also tried to just plug in Laurent series expanded at $a$ in $f'/f$ and do calculation to eliminate some coefficients with negative indices but calculation would result infinite system of linear equations so I gave up doing it. Therefore, I'm stuck and please give me some hint.
Asked
Active
Viewed 20 times
1
YD55
- 169