This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 "branchcut") 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 pathindepen  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[10]:= c1 = 8 0.5, 0.5 < ; c2 = 8 1., 1. < ; GraphicsRow @ 8 Show @ ParametricPlot @8 c1 + 8 Sin @ t D , Cos @ t D< , c2...
View
Full
Document
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
 HussainQassem

Click to edit the document details