Lecture 21 Lecture Notes

Lecture 21 Lecture Notes - with . The integral vanishes!...

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Recap: a solution to for the same boundary conditions is given by Looks easy enough, but: Integrand has poles at 7.2.3 Evaluation of Green function integral =>Embedding into complex parameter space: Evaluate via Residue Theorem retarded G. funct. advanced G.funct. all paths possible and valid, but leading to different structure of boundary conditions - different Green functions - homogeneous wave equation! => differ only by solutions of Residue theorem (simplified): - Analytic function f(z) on z - Simple closed path C in complex plane Given Then Where the sum goes over all points a_k enclosed by the path C. Note: mathematical positive direction for path C! Lec 21 - 2. Mar March 01 09 9:54 PM Lecture Notes Page 1
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Residues: Define for for => for t > 0 close loop with lower circle Closure of path: For simple poles at point "a" we have or (acquieres minus sign from path direction! Send cut off points along axis to infinity Find situations such t hat in the integral ==> for example: integrand falls off faster than
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Unformatted text preview: with . The integral vanishes! Closure in upper half possible for t<0! Lecture Notes Page 2 7.2.4 Retarded Green function The path of integration is chosen such that the singularity are below the curve approximating the real axis, and is closed in each case so that integral over the arch vanishes. from integration with clockwise orientation Lecture Notes Page 3 Note: Consider solutions of If for then automatically this solutions has no incoming waves! Proof: Since in the domain We have Lecture Notes Page 4 7.2.5 Advanced Green function Symmetry if for no outgoing waves! for => Difference between solution obtained with retarded and with advanced Green function are just the outgoing and incoming waves! outgoing wave of retarded solution destructive interference for advanced solution incoming wave of advanced solution source and especially Lecture Notes Page 5...
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This note was uploaded on 07/28/2009 for the course PHYS 441B taught by Professor Norbertlutkenhaus during the Winter '09 term at Waterloo.

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Lecture 21 Lecture Notes - with . The integral vanishes!...

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