math185-hw4 - MATH 185 COMPLEX ANALYSIS FALL 2007\/08 PROBLEM SET 4 We write C = C{0 We say that f is identically zero on denoted f 0 if f(z = 0 for all

# math185-hw4 - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 4 We write C × = C \{ 0 } . We say that f is identically zero on Ω, denoted f 0, if f ( z ) = 0 for all z Ω. When we write f ( z ) > 0 for a complex function f , it is implicit that f ( z ) R (likewise for < , , , and if 0 is replaced by any other real number). You may use without proof any results that had been proved in the lectures. 1. Prove or disprove. Given any entire function f : C C , there exist functions g, h : C C such that (i) g and h are both entire functions, (ii) f ( z ) = g ( z ) + ih ( z ) for all z C , (iii) g ( x ) R and h ( x ) R for all x R . 2. (a) Let J be defined by the power series J ( z ) = X n =0 ( - 1) n ( n !) 2 z 2 2 n . Prove that z 2 J 00 ( z ) + zJ 0 ( z ) + z 2 J ( z ) = 0 . State which theorem(s) you have used here. For what values of z is this valid? (b) More generally, for any k N ∪ { 0 } , let J k be defined by the power series J k ( z ) = X n =0 ( - 1) n n !( n + k )! z 2 2 n + k . Prove that z 2 J 00 k ( z ) + zJ 0 k ( z ) + ( z 2 - k 2 ) J k ( z ) = 0 . For what values of z is this valid? 3. (a) For i = 1 , 2 and j = 1 , 2 , 3 , 4, determine the value of Z Γ j f i where f i is defined by f 1 : C C , f 1 ( z ) = z 3 , f 2 : C C , f 2 ( z ) = z ; and Γ j is defined by z 1 : [0 , 1] C , z 1 ( t ) = 1 + it, z 2 : [0 , 1] C , z 2 ( t ) = e - iπt , z 3 : [0 , 1] C , z 3 ( t ) = e iπt , z 4 : [0 , 1] C , z 4 ( t ) = 1 + it + t 2 . Date : October 17, 2007 (Version 1.2); posted: October 11, 2007; due: October 19, 2007.

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