- #1
binbagsss
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Homework Statement
See attached.
The solution of part e) is ##C=4\psi(a)##
I am looking at part e, the answer to part d being that the principal parts around the poles ##z=0## and ##z=-a## are the same.
Homework Equations
The Attempt at a Solution
[/B]
Since we already know the negative powers of ##z## have the same expansions, and ##C## corresponds to the ##z^0## term, ##f_a(z)^2## about ##z=0## gives ##\frac{4}{z^2}+4\psi(a)z^2+4\psi(a)## and so the relevant term is ##4\psi(a)##.
Looking at the expansion of ##f_a(z)^2## about ##z=-a## there is no ##z^0## term so I conclude ##C=4\psi(a)##.
QUESTION
- This doesn't really seem like a proper approach, i.e to break it down to considering the expansions of ##f_a(z)## about ##z=0 ## and ##z=-a## separately, whereas I am considering the RHS as a function over the entire complex plane . ( if I wasn't considering the RHS or LHS over the entire complex plane then the theorem to give the additive constant ##C## does not work, so I don't really see how you can break it down on either the LHS or RHS to consider only an expansion about a single pole ? )
Thanks in advance.