math185-hw10 - MATH 185 COMPLEX ANALYSIS FALL 2007\/08 PROBLEM SET 10 For a C 0 < r < R we write A(a r R:={z C | r < |z a| < R We also write C = A(0 r

# math185-hw10 - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

• Notes
• mjg68
• 1

This preview shows page 1 out of 1 page.

MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 10 For a C , 0 < r < R , we write A ( a ; r, R ) := { z C | r < | z - a | < R } . We also write C × = A (0; r, ) and D * ( a, r ) = A ( a ; 0 , r ). The number of zeroes, counted with multiplicity, of f in Ω is denoted Z f (Ω). You may use without proof any results that had been proved in the lectures. 1. Let the Laurent expansion of cot( πz ) on A (0; 1 , 2) be cot( πz ) = X n = -∞ a n z n . Compute a n for n < 0. 2. (a) For n = 0 , 1 , 2 , . . . , compute 1 2 πi Z Γ n dz z 3 sin z where Γ n is the circle ∂D (0 , r n ) traversed once counter-clockwise and r n = ( n + 1 2 ) π . (b) Evaluate 1 2 πi Z Γ z 11 12 z 12 - 4 z 9 + 2 z 6 - 4 z 3 + 1 dz where Γ = ∂D (0 , 1) traversed once counter-clockwise. 3. Determine Z f i i ) for i = a, b, c . (a) f a ( z ) = z 87 + 36 z 57 + 71 z 4 + z 3 - z + 1 and Ω a = D (0 , 1). (b) f b ( z ) = z 87 + 36 z 57 + 71 z 4 + z 3 - z + 1 and Ω b = D (0 , 2). (c) f c ( z ) = 2 z 5 - 6 z 2 + z + 1 and Ω c = A (0; 1 , 2). 4. (a) Prove the following form of Morera’s Theorem:

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

Want to read the whole page?

Unformatted text preview: ⊆ C be an open set. Let f be continuous on Ω. Show that if Z ∂R f = 0 for every ∂R ⊂ Ω, boundary of a rectangle R ⊂ C (note that Ω may not be simply connected and R may not be contained in Ω), then f has an antiderivative on Ω, ie. there exists an analytic F on Ω such that F = f . (b) Does the following function have an antiderivative on A (0;4 , ∞ )? z ( z-1)( z-2)( z-3) (c) Does the following function have an antiderivative on A (0;4 , ∞ )? z 2 ( z-1)( z-2)( z-3) Date : December 6, 2007 (Version 1.0); posted: December 6, 2007; due: December 10, 2007. 1...
View Full Document

• Fall '07
• Lim
• Math, following function

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern