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
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01
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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