Periods of Jacobi Elliptic functions

[binbagsss]

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

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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.

 

[binbagsss]

bump.