Recap:a solution tofor the same boundary conditions is given byLooks easy enough, but: Integrand has poles at7.2.3 Evaluation of Green function integral=>Embedding into complex parameter space:Evaluate via Residue Theoremretarded G. funct.advanced G.funct.all paths possible and valid, but leading todifferent 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 planeGivenThenWhere the sum goes over all points a_k enclosed by the path C.Note: mathematical positive direction for path C!Lec 21 - 2. MarMarch‐01‐099:54 PMLecture Notes Page 1
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Residues:Defineforfor=> for t > 0 close loop with lower circleClosure of path:For simple poles at point "a" we haveor(acquieres minus signfrom path direction!Send cut‐off points along axis to infinityFind situations such t hat in the integral==> for example: integrand falls off faster than