math185-hw4sol - MATH 185 COMPLEX ANALYSIS FALL 2007\/08 PROBLEM SET 4 SOLUTIONS We write C = C{0 We say that f is identically zero on denoted f 0 if f(z

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

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 4 SOLUTIONS 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 . 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 + n where β n , γ n R for all n N ∪ { 0 } . 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 ∪ { 0 } . So 0 lim sup n →∞ n p | β n | ≤ lim sup 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 ∪ { 0 } , it is clear that g ( x ) R and h ( x ) R for all x R .
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