Elliptic Functions, same principal parts, finding additive C

[binbagsss]

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.

 

[mighty2000]

Thank you for your post. Your approach to solving part e) of the problem is correct. It is important to consider the expansion of the function about each individual pole separately, as the function may have different expansions at different poles. In this case, the function has the same expansion at both poles, which allows us to conclude that the constant term, C, is equal to 4ψ(a).

It is also important to consider the function over the entire complex plane, as this allows us to understand its behavior and properties. However, in this case, we are specifically asked to determine the value of C, which only requires us to consider the expansions at the two poles.

I hope this helps clarify your approach. Keep up the good work in your studies!Scientist