math185-hw9sol

math185-hw9sol - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 9 SOLUTIONS For a C , r > 0, we write D * ( a,r ) := { z C | < | z- a | < r } . We write C = C \{ } . You may use without proof any results that had been proved in the lectures. 1. Evaluate the integral Z i f i for i = a,b . (a) f a : C C is given by f a ( z ) = e e 1 z and a is the boundary D (0 , 2) traversed once counter-clockwise. Solution. f a is analytic in C and its Laurent expansion about z = 0 may be obtained as follows: e e 1 z = 1 + e 1 z + 1 2! e 2 z + + 1 n ! e n z + = 1 + 1 + 1 z + 1 2! 1 z 2 + + 1 2! " 1 + 2 z + 1 2! 2 z 2 + # + + 1 n ! 1 + n z + 1 2! n z 2 + + . Observe that the coefficient of the term z- 1 is simply 0 + 1 + 2 2! + 3 3! + + n n ! + = 1 + 1 1! + 1 2! + + 1 ( n- 1)! + = e. By the residue theorem Z a f a = 2 i Res( f a ;0)Ind( a ;0) = 2 ei....
View Full Document

Page1 / 3

math185-hw9sol - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online