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Unformatted text preview: Complex Analysis Fall 2007 Homework 8: Solutions 2.R.1 (a) Since sin z is entire, its integral around any closed curve is zero by Cauchy’s Theorem. (b) Since sin z is entire and γ has a winding number of 1 about 0, the Cauchy Integral Formula immediately gives 0 = sin 0 = 1 2 πi Z γ sin z z dz so the integral is zero. (c) Since sin z is entire and γ has a winding number of 1 about 0, the Cauchy Integral Formula immediately gives 1 = cos 0 = d dz (sin z ) z = 0 = 1 2 πi Z γ sin z z 2 dz so the integral is 2 πi . (d) Since f ( z ) = sin e z is entire with f ( z ) = e z cos e z , the Cauchy Integral Formula imme- diately gives cos 1 = e cos e = 1 2 πi Z | z | =1 sin e z z 2 dz so the integral is 2 π (cos 1) i . 2.R.3 See the textbook’s solution. 2.R.4 (a) Since deg z 2 P ( z ) = 2 + deg P ( z ) ≤ deg Q ( z ) we find that lim z →∞ z 2 P ( z ) Q ( z ) = a for some a ∈ C (i.e. we are not in the case when the limit is infinite). Choose R > 0 so that | z 2 P ( z ) /Q ( z )...
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This note was uploaded on 02/29/2012 for the course MATH 4364 taught by Professor Ryandaileda during the Fall '11 term at Trinity University.
- Fall '11