hw4 - Mathematics 185 – Intro to Complex Analysis Fall...

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Unformatted text preview: Mathematics 185 – Intro to Complex Analysis Fall 2009 – M. Christ Solutions to Selecta from Problem Set 4 ( x.R.n denotes the n-th review problem for Chapter x .) 1.R.21 Let D = { z : | z- 1 | < 1 } and let f : D → C be analytic and satisfy f ( z ) = 1 /z and f (1) = 0. Show that f ( z ) = log( z ). Solution. Let L ( z ) denote the unique logarithm of z with argument in (- π/ 2 ,π/ 2), for z ∈ C \ (-∞ , 0]. I will show more precisely, and less ambiguously, that f ( z ) = L ( z ) for all z ∈ D . Consider g ( z ) = f ( z )- L ( z ) for z ∈ D . Then g ( z ) ≡ 0. Since D is connected, g is constant. Therefore f ( z )- L ( z ) ≡ f (1)- L (1) = 0- 0 = 0. 1.R.26. Let g : A → C be an analytic function on any open set A . Let B = { z ∈ A : g ( z ) 6 = 0 } . Show that B is open, and that 1 /g is an analytic function on B . Solution. Since g is analytic, it is continuous. Given any z ∈ B , define ε = | g ( z ) | / 2. There exists δ > 0 such that | w- z | < δ ⇒ | g ( w )- g ( z ) | < ε . By the triangle inequality, for any such w , | g ( w ) | ≥ | g ( z ) | - | g ( w )- g ( z ) | ≥ 1 2 | g ( z ) | . Therefore B contains the open disk D ( z,δ ). By definition of an open set, B is open. Define G ≡ g , but with domain restricted to B . Then G is analytic. Define h ( w ) = 1 /w for all w ∈ C \ { } . Then h is analytic, and its domain contains the range of G . Since 1 /g = h ◦ G is the composition of two analytic functions, it is also analytic. 1.R.28 Let f : A → C be analytic and assume that f ( z ) 6 = 0 for every z ∈ A . Prove that { Re( f ( z )) : z ∈ A } is an open subset of R . Solution. Let B = { Re( f ( z )) : z ∈ A } . In order to prove that B is an open subset of R , we must show that for any s ∈ B , there exists r > 0 such that ( s- r,s + r ) ⊂ B . Choose z ∈ A satisfying Re( f ( z )) = s . By the inverse function theorem, since f ( z ) 6 = 0, f ( A ) contains some disk D = D ( f ( z ) ,r ) of positive radius r . Therefore B contains every s ∈ R which is the real part of some element of D . Write f ( z...
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This note was uploaded on 10/10/2009 for the course MATH 185 taught by Professor Lim during the Spring '07 term at Berkeley.

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hw4 - Mathematics 185 – Intro to Complex Analysis Fall...

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