CalcIVComplexSol

CalcIVComplexSol - Solutions to sample questions for...

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Solutions to sample questions for complex integration 1. Let C be the curve z ( t ) = t 2 + it 3 for 0 t 1 , and let f ( z ) = e πiz . Compute R C f ( z ) dz . We see right away that f ( z ) is holomorphic (since it’s the composition of functions we know to holomorphic), and thus we can use the fundamental theorem of calculus. The curve starts at 0 and ends at 1 + i , so we have Z C f ( z ) dz = ± - i π e πiz ² 1+ i 0 = i π ( 1 + e - π ) . 2. Let C be the line segment from i to 1, and let f ( z ) = z · z . Compute R C f ( z ) dz . We have no reason to believe that f ( z ) is holomorphic (and, in fact, it isn’t as can be seen from the Cauchy-Riemann equations). So we compute the integral directly. We can parametrize C by z ( t ) = t + (1 - t ) i for 0 t 1. Then z 0 ( t ) = 1 - i and f ( z ( t )) = 2 t 2 - 2 t + 1, after a little algebra. Thus Z C f ( z ) dz = Z 1 0 ( 2 t 2 - 2 t + 1 ) (1 - i ) dt = (1 - i ) ± 2 3 t 3 - t 2 + t ² 1 0 = 2 3 - 2 3 i 3. Let C be the curve consisting of the line segment from
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This note was uploaded on 07/25/2008 for the course MATH V1202 taught by Professor Neel during the Spring '07 term at Columbia.

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CalcIVComplexSol - Solutions to sample questions for...

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