Transcription of Residues and Contour Integration Problems
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Residues and Contour Integration Problems
Residues and Contour Integration ProblemsClassify the singularity off(z)at the indicated (z) = cot(z) atz= 0. Ans. Simple The test for a simple pole atz= 0 is that limz 0zcot(z)exists and is not 0. We can use L H opital s rule:limz 0zcot(z) = limz 0zcos(z)sin(z)= limz 0cos(z) zsin(z)cos(z)= the singularity is a simple (z) =1+cos(z)(z )2atz= . Ans. Power series is the simplest way to do this. We can expandcos(z) in a Taylor series aboutz= . To do so, use the trig identitycos(z) = cos(z ). Next, expand 1 cos(z ) in a power seriesinz :1 + cos(z) = 1 cos(z ) =12!(z )2 14!(z )4+ From this, we get1 + cos(z)(z )2=(z )2(12 14!(z )2+ )(z )2=12 14!(z )2+ ,which is the Laurent series for1+cos(z)(z )2.
Residues and Contour Integration Problems Classify the singularity of f(z) at the indicated point. 1. f(z) = cot(z) at z= 0. Ans. Simple pole. Solution.
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