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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