- #1
binbagsss
- 1,326
- 12
Homework Statement
I have that ##(\psi(z)-e_j)^{1/2}=e^{\frac{-n_jz}{2}}\frac{\sigma(z+\frac{w_j}{2})}{\sigma(\frac{w_j}{2})\sigma(z)}##
has period ##w_i## if ##i=j##
and period ##2w_i## if ##i\neq j##
where ##i,j=1,2,3## and ##w_3=w_1+w_2## (*)
where ##e_j=\psi(\frac{w_j}{2})##
I have the following (two) identities:
##\sigma(z+w_j)=-\sigma(z)e^{n_j(z+\frac{w_j}{2})}## (1)
From which we can get:
##\sigma(z+\frac{w_j}{2})=-e^{n_j z}\sigma(z-\frac{w_j}{2})## (2)
With the above information I need to deduce that:
##S(z)## has period lattice with ##(2w_1,w_2)##
##C(z)## has period lattice with ##(2w_1,w_1+w_2)##
##D(z)## has period lattice with ##(w_1,2w_2)##
where ##S(z), C(z), D(z)## are given as attached:
Homework Equations
see above, see below, look up, look down, look all around.
The Attempt at a Solution
[/B]
My book says this is straightforward with the result (*) stated above. But I can't see it for ##C(z) ## and ##D(z)##...
For example,
##C(z)## numerator, using (*), has periods ##(w_1,2w_2)##, whilst the denominator has periods ##(2w_1,w_2)##..
So I can't see how a obvious conclusion can be made? (Similarly for ##D(z)##)...
All I can see to do is work through a tonne of algebra, each period in turn, using (1) and (2), however then I'm not making use of (*), from which my book says the result simply follows, so I don't think I should be needing to do this anyway...
Many thanks in advance.