math185f08-hw4sol - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 4 SOLUTIONS For z ∈ C , recall that the argument of z , denoted arg( z ), is any θ ∈ R such that z = | z | e iθ . Note that arg( z ) is only defined modulo 2 π . 1. This problem shows that the conditions in Theorem 2.8 in the lectures cannot be omitted. (a) Let f : C → C be defined by f ( z ) = ( exp(- z- 4 ) z 6 = 0 , z = 0 . Show that f satisfies the Cauchy-Riemann equation at all z ∈ C but f is not analytic on C . Why doesn’t this contradict the second part of Theorem 2.8 ? Solution. For z = x + iy ∈ C × , it follows from chain rule that f x ( z ) = 4 z- 5 exp(- z- 4 ) , f y ( z ) = 4 iz- 5 exp(- z- 4 ) . Hence the Cauchy-Riemann equation f x ( z ) = if y ( z ) is satisfied for all z ∈ C × . To compute f x (0) and f y (0), we need to use the definition of partial derivatives. f x (0) = lim ξ → ,ξ ∈ R f (0 + ξ )- f (0) ξ = lim ξ → ,ξ ∈ R exp(- ξ- 4 )- ξ = lim s →∞ ,s ∈ R s exp( s 4 ) = 0 . This follows from setting s = 1 /ξ and using the exponential inequality e x ≥ 1+ x for a real variable x to get ≤ s exp( s 4 ) ≤ s 1 + s 4 = 1 s- 1 + s 3 → as s → ∞ . The same argument together with the observation that i 4 = 1 gives f y (0) = lim η → ,η ∈ R f (0 + iη )- f (0) η = lim η → ,η ∈ R exp(- η- 4 )- η = lim s →∞ ,s ∈ R s exp( s 4 ) = 0 . Hence the Cauchy-Riemann equation is also satisfied at z = 0 since f x (0) = 0 = if y (0) . Date : October 24, 2008 (Version 1.0). 1 However f is not analytic on C since it is not analytic at z = 0. In fact, it is not even continuous at z = 0. To see this, write z = r (cos θ + i sin θ ) and use De Moivre’s formula to get exp(- z- 4 ) = exp(- r- 4 cos4 θ + ir 4 sin4 θ ) = exp(- r- 4 cos4 θ )exp( ir 4 sin4 θ ); if we consider the sequence z n := 1 n cos π 8 + i sin π 8 , n ∈ N , then z n → 0 but f ( z n ) = exp(- z- 4 n ) = exp( in- 4 ) 9 since | exp( in- 4 ) | = 1 for all n ∈ N . So f is not continuous at 0....
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This note was uploaded on 06/15/2009 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

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math185f08-hw4sol - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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