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Unformatted text preview: Homework 5 Stephen Taylor June 6, 2005 Page 44 16. Find an open connected set G ⊂ C and two continuous functions f and g defined on G such that f ( z ) 2 = g ( z ) 2 = 1 z 2 for all z ∈ G . Can you make G maximal? Are f and g analytic? We define f ( z ) ≡ p 1 z 2 g ( z ) ≡  p 1 z 2 which we see satisfy the desired equation f ( z ) 2 = g ( z ) 2 = 1 z 2 We note that since ∂f ∂ ¯ z = ∂g ∂ ¯ z = 0 both f and g are analytic. Moreover, the definition of f ( z ) is given by f ( z ) = p 1 z 2 ≡ exp 1 2 log 1 z 2 which is defined everywhere except z = 1 , 1. So we may defined f, g over any simply connected open set of the complex plane not including ± 1. 17. Give the principal branch of √ 1 z . √ 1 z = (1 z ) (1 / 2) = exp 1 2 Log (1 z ) = exp 1 2 Log (1 x iy ) = exp 1 2 ln p  (1 x ) 2 + y 2  + i arctan y x 1 = p (1 x ) 2 + y 2 (cos θ + i sin θ ) where θ = arctan y x 1 1 Page 54 55 1. Find the image of { z : Re z < ,  Im z  < π } under the exponential function.under the exponential function....
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 Three '09
 Cong
 Complex number, Stephen Taylor, open unit disk, exponential maps

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