math185f09-hw7sol

# math185f09-hw7sol - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 7 SOLUTIONS 1. Let β ∈ C . (a) Show that for all n = 0 , 1 , 2 ,..., β n n ! 2 = 1 2 πi Z ∂D (0 , 1) β n e βz n ! z n +1 dz. Solution. Applying generalized Cauchy’s integral formula, we get 1 2 πi Z ∂D (0 , 1) β n e βz n ! z n +1 dz = β n n ! " 1 2 πi Z ∂D (0 , 1) e βz z n +1 dz # = β n n ! × 1 n ! d n dz n e βz z =0 = β n n ! 2 . (b) Show that ∞ X n =0 β n n ! 2 = 1 2 π Z 2 π e 2 β cos θ dθ. [ Hint : Consider power series expansion of e β/z and apply (a) on z- 1 e β ( z +1 /z ) .] Solution. Note that e β/z = ∞ X n =0 β n n ! z n . Multiplying by e βz and dividing by z , we get 1 z e β ( z +1 /z ) = ∞ X n =0 β n e βz n ! z n +1 . Integrating about ∂D (0 , 1) and using (a), we get 1 2 πi Z ∂D (0 , 1) 1 z e β ( z +1 /z ) dz = ∞ X n =0 β n n ! " 1 2 πi Z ∂D (0 , 1) e βz z n +1 dz # = ∞ X n =0 β n n ! 2 Evaluating the line integral about the path z : [0 , 2 π ] → C , z ( θ ) = e iθ and noting that e iθ + e- iθ = 2cos θ we get 1 2 πi Z ∂D (0 , 1) 1 z e β ( z +1 /z ) dz = 1 2 πi Z 2 π e 2 β cos θ dθ. 2. Let f : C → C be an entire function. Let a ∈ R be an arbitrary constant. (a) Show that if Re f ( z ) ≤ a for all z ∈ C , then f is constant. (b) Show that if Re f ( z ) ≥ a for all z ∈ C , then f is constant. (c) Show that if [Re f ( z )] 2 ≤ [Im f ( z )] 2 for all z ∈ C , then f is constant. (d) Show that if [Re f ( z )] 2 ≥ [Im f ( z )] 2 for all z ∈ C , then f is constant. (e) Suppose h is another entire functions and suppose there exists an a ∈ R , a > 0, such that Re f ( z ) ≤ a Re h ( z ) for all z ∈ C . Show that there exist α,β ∈ C such that f ( z ) = αh ( z ) + β for all z ∈ C . Date : November 14, 2009 (Version 1.0). 1 [ Hint : if f and g are both entire, then so are f ◦ g and g ◦ f ; find an appropriate g so that you may apply Liouville’s theorem.] Solution. Note that e x is a monotone increasing function on R ....
View Full Document

## This note was uploaded on 02/17/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

### Page1 / 5

math185f09-hw7sol - MATH 185: COMPLEX ANALYSIS FALL 2009/10...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online