185hw5.pdf - Math 185 Homework 5 \u2013 Due 10\/12 Peter Koroteev 1 Compute the following integrals Z dz 2 z \u22121 |z|=2 Z sin z dz z |z|=1 Z cos(z 2 dz z

185hw5.pdf - Math 185 Homework 5 – Due 10/12 Peter...

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Math 185 Homework 5 – Due 10/12 Peter Koroteev 1. Compute the following integrals Z | z | =2 dz z 2 - 1 , Z | z | =1 sin z z dz , Z | z | =1 cos( z 2 ) z dz , where the contours represent circles with a counter-clockwise direction. 2. Compute Z | z | = r x dz , where the contour represents a circle of radius r with a counter-clockwise direction, in two ways: first, using an explicit parameterization, and, second, by observing that x = z z 2 = 1 2 z + r 2 z on the circle. 3. Suppose that f ( z ) is holomorphic on a closed curve
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Unformatted text preview: γ (homomorphic in an open set which contains γ ). Show that Z γ f( z )f (z ) dz ,is purely imaginary. 4. Compute Z | z | = ρ | dz| |z-a | 2 ,Z | z | =ρ | dz| | z-a | 4 , assuming that | a| 6 =ρ . 5. Prove that if fis holomorphic in C and | f( z ) | < |z | n for some integer nand all sufficiently large | z| reduces to a polynomial....
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  • Fall '07
  • Lim
  • Topology, Metric space, Conic section, The Contours, Peter Koroteev

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