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Periods of Jacobi Elliptic functions

  1. Apr 1, 2017 #1
    1. The problem statement, all variables and given/known data

    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:

    attachhere.png

    2. Relevant equations

    see above, see below, look up, look down, look all around.

    3. The attempt at a solution



    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.
     
  2. jcsd
  3. Apr 6, 2017 #2
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