V log z z 1 for all z ω vi if z 1 z 2 z 1 z 2 ω

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(v) log ( z ) = z 1 for all z Ω . (vi) If z 1 , z 2 , z 1 z 2 Ω , then log z 1 z 2 = log z 1 + log z 2 + 2 nπi for some n Z . Lemma 2.12. There does not exist a continuous function L : C \ { 0 } → C such that exp( L ( z )) = z . Exercise 2.13. We use the notation of Definition 2.10. Show that we cannot choose θ 0 so that log z 1 z 2 = log z 1 + log z 2 for all z 1 , z 2 , z 1 z 2 Ω . In Analysis I, you saw that the easiest way to define x α when x and α are real and x> 0 is to write x α = exp( α log x ). Definition 2.14. We use the notation of Definition 2.10. Suppose α C . We define the map z mapsto→ z α on Ω by z α = exp( α log z ) . We call the resulting function a branch of z α . 6
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Lemma 2.15. If we define p α : Ω C by p α ( z ) = z α as in Definition 2.14, then p α is analytic on Ω . We have p α ( z ) = αp α 1 ( z ) . If α is real, r> 0 and 2 π + θ 0 >θ>θ 0 then p α ( z ) = r α exp iαθ where r α has its traditional meaning. Except in the simplest circumstances, it is probably best to deal with z α by rewriting it as exp( α log z ). If 0 0 > 2 π , it is traditional to refer to the function defined by log re = log r + for r> 0 and 2 π + θ 0 >θ>θ 0 as the principal branch of the logarithm (with a similar convention for the associated powers). This has the same effect and utility as my referring to myself as the King of Siam. 3 Conformal mapping We start with our definition of a conformal map. Definition 3.1. Let Ω and Γ be open subsets of C . We say that f : Ω Γ is a conformal map if f is bijective and analytic and f never vanishes. In more advanced work it is shown that, if f is bijective and analytic, then f never vanishes. The phrase ‘and f never vanishes’ can then be omitted from the definition. Lemma 3.2. Let Ω and Γ be open subsets of C . If f : Ω Γ is conformal, then f 1 : Γ Ω is analytic. We have ( f 1 ) ( w ) = 1 f ( f 1 ( w )) , so f 1 is also conformal. Exercise 3.3. We say that open subsets Ω and Γ of C are conformally equiv- alent if there exists a conformal map f : Ω Γ . Show that conformal equivalence is an equivalence relation. The reader is warned that some mathematicians use definitions of confor- mal mapping which are not equivalent to ours. (The most common change is to drop the condition that f is bijective but to continue to insist that f is 7
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never zero.) Sometimes people use conformal simply to mean angle preserv- ing, so you must be prepared to be asked ‘Show that an analytic map with non-zero derivative is conformal’. So far as 1B examinations are concerned, we are chiefly interested in the following conformal maps. (i) z mapsto→ z + a . Translation. Takes C to C . (ii) z mapsto→ e z where θ is real. Rotation. Takes C to C . (iii) z mapsto→ λz where λ is real and λ> 0. Dilation (scaling). Takes C to C .
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