math185-hw4sol - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 4 SOLUTIONS We write C × = C \{ } . 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 . Solution. By Theorem 4.3 in the lectures, f has a power series representation f ( z ) = ∞ X n =0 a n z n for all z ∈ C (ie. the radius of convergence of the rhs is ∞ ). Let a n = β n + iγ n where β n ,γ n ∈ R for all n ∈ N ∪ { } . We define g,h by g ( z ) = ∞ X n =0 β n z n and h ( z ) = ∞ X n =0 γ n z n . Note that | β n | ≤ | a n | for all n ∈ N ∪ { } . So ≤ limsup n →∞ n p | β n | ≤ limsup n →∞ n p | a n | = 0 , and the series defining g has an infinite radius of convergence. Likewise, the series defining h has an infinite radius of convergence. Hence g and h both entire functions. Since the series defining f , g , and h all have infinite radii of convergence, the following equation is valid for all z ∈ C : ∞ X n =0 a n z n = ∞ X n =0 β n z n + i ∞ X n =0 γ n z n (note that this is not true in general — see Chapter 2 , Exercise 10 in the textbook). Hence we have f ( z ) = g ( z ) + ih ( z ) for all z ∈ C . Since β n ,γ n ∈ R for all n ∈ N ∪ { } , it is clear that g ( x ) ∈ R and h ( x ) ∈ R for all x ∈ R ....
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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 University of California, Berkeley.

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math185-hw4sol - MATH 185: COMPLEX ANALYSIS FALL 2007/08...

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