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Unformatted text preview: A Summary of Contour Integration vs 0.2, 25/1/08 ' Niels Walet, University of Manchester In this document I will summarise some very simple results of contour integration 1. The Basics The key word linked to contour integration is "analyticity" or the absence thereof: A function is called analytic in a region R in the complex plane iff all the derivatives of the function (1st, 2nd, ....) exist for every point inside R . This means that the Taylor series ( 1 ) f H z L = n = ¥ f H n L H c L H z- c L n n ! exists for every point c inside R . In most cases we are actually interested in functions that are not analytic; if this only happens at isolated points (i.e., we don’t consider a "line of singularities", usually called a "cut" or "branch-cut") we can still expand the function in a Laurent series ( 2 ) f H z L = n =-¥ ¥ f H n L H c L H z- c L n n ! How we obtain the coefficients f H n L H c L is closely linked to the problem of contour integration. 2. Contour Integration Let us look at the effects of integrating the powers of z along a line in the complex plane (note that we implicitely assume that the answer is independent of the position of the line, and only depends on beginning and end!) ( 3 ) z z 1 z n z , n ˛ Z . We know how to integrate powers, so apart from the case n = - 1, we get ( 4 ) z z 1 z n z = B 1 n + 1 z n + 1 F z 1 z , n „ - 1, z z 1 z- 1 z = @ log z D z 1 z , n = - 1. Contour equation addresses the fact that the first of these integrals is indeed right, and as we expect for path-indepen - dence, goes to zero as we look at a closed path, but the second integral actually does depend on the path of integration: ( 5 ) Use z = r e Φ ; log H z L = log H r L + Φ . We have two options: the contour (the technical word for the closed path) either encloses the origin where the singular- ity resides or not: In:= c1 = 8 0.5, 0.5 < ; c2 = 8 1., 1. < ; GraphicsRow @ 8 Show @ ParametricPlot @8 c1 + 8 Sin @ t D , Cos @ t D< , c2...
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This note was uploaded on 10/29/2011 for the course ELEC 201 taught by Professor Hussainqassem during the Spring '11 term at Qatar University.
- Spring '11