HW9.EGM4313

# HW9.EGM4313 - Page626, Problem 21: e z = 4 − 3i = 5e i (...

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Unformatted text preview: Page626, Problem 21: e z = 4 − 3i = 5e i ( arctan ( −3 / 4 ) + 2πn ) ⇒ z = ln ( 5) + i ( arctan( − 3 / 4) + 2πn ) Page 633, problem 25: (1 + i ) 1−i () Page 645, problem 22: 2i ∫ sin ( z )dz = − cos( 2i ) + cos( 0) = 1 − cosh( 2) 0 Let z=iy => dz=idy. 2i 2 2 ( 2 ) 1 −y y ∫ sin ( z )dz = ∫ i sin ( iy )dy = ∫ 2 e − e dy = ∫ − sinh( y )dy = 1 − cosh( 2) 0 0 0 0 Page 645, problem 26: z is NOT analytic, therefore can only use method #2. Let z=x+iy => z=x+ix2 on C => dz = (1+2ix)dx and z = x - i x2 on C. ∫ zdz = C 1 1 2 ∫ ( x − ix )(1 + 2ix )dx = ∫ ( x + 2 x ) + i( x )dx = 3 i 2 3 −1 −1 Page 653, problem 3: e z2 / 2 () π π = e ( 1−i ) ( ln ( 1+i ) ) = e ( 1−i ) ( ln ( 2 )+i ( π / 4 ) ) = 2e π / 4 cos − ln 2 + i sin − ln 2 4 4 z is analytic everywhere ∫ e C 2 /2 dz = 0. 2 Page 653, problem 4: 2π 1 1 1 1 z = e iθ ⇒ dz = ie iθ dθ , = −iθ = e iθ ⇒ ∫ dz = ∫ ie 2iθ dθ = 0. f ( z ) = not analytic. Let ze z z C 0 ...
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## This note was uploaded on 09/05/2011 for the course EGM 4313 taught by Professor Mei during the Spring '08 term at University of Florida.

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HW9.EGM4313 - Page626, Problem 21: e z = 4 − 3i = 5e i (...

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