# Home_5 - Homework 5 Stephen Taylor June 6 2005 Find an open...

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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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Home_5 - Homework 5 Stephen Taylor June 6 2005 Find an open...

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