#### You've reached the end of your free preview.

Want to read the whole page?

**Unformatted text preview: **at a . 3. Let Ω ⊆ C be a region. Let a ∈ Ω and f : Ω \{ a } → C be a function with an isolated singularity at a . Suppose for some m ∈ N and ε > 0, Re f ( z ) ≤ -m log | z-a | for all z ∈ D * ( a,ε ). Show that a is a removable singularity of f . 4. Let f : C × → C be analytic on C × with a pole of order 1 at 0. Show that if f ( z ) ∈ R for all | z | = 1, then for some α ∈ C × and β ∈ R , f ( z ) = αz + α 1 z + β for all z ∈ C × . 5. Let f : D * (0 , 1) → C be analytic. Show that if | f ( z ) | ≤ log 1 | z | for all z ∈ D * (0 , 1), then f ≡ 0. Date : November 15, 2007 (Version 1.0); posted: November 15, 2007; due: November 21, 2007. 1...

View
Full Document

- Fall '07
- Lim
- Math, Essential singularity, singularity, isolated singularity