math185-hw4

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 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 ) =
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This note was uploaded on 08/01/2008 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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

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