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Typo in DedekindEta function definition?


Hello all,

There's something puzzling me about Mathematica's definition of the
Dedekind eta function in terms of the Weierstrass invariants g2, g3.
Recall that the function WeierstrassInvariants[{w, w' }] involve the
half-periods {w, w'} and KleinInvariantJ[z] and DedekindEta[z] the
half-period ratio z = w' /w. To illustrate, (Notes: For brevity, values
have been rounded off. The symbol "i" is the imaginary unit.):

In[1]: = {g2,g3} = N[WeierstrassInvariants[{3, 5i}], 30]
Out[1] = {0.10089, 0.00601}

In[2]:= p1 = N[KleinInvariantJ[5i/3], 30]
Out[2] = 20.86892

In[3]:= p2 = g2^3/(g2^3-27g3^2)
Out[3] = 20.86892

As expected, p1 and p2 are equal. However,

In[4]:= r1 = N[(2Pi)^12 DedekindEta[5i/3]^24, 30]
Out[4] = 107137.63536

In[5]:= r2 = g2^3-27g3^2
Out[5] = 0.0000492

Shouldn't r1 and r2 be equal? After some experimentation, I found they
can be if the equality is modified to r1 = r2 (2w)^12 where "w" is the
first half-period.

In[6]:= r2 (6)^12
Out[6] = 107137.63536

and we now get the same value.  In the help section, the Klein
invariant J[z] is defined as g2^3 /(g2^3-27g3^2) which as we saw
worked.  But the Dedekind eta, or n[z], is defined as satisfying d =
(2Pi)^12 n[z]^24 where d is the discriminant and given in terms of the
Weierstrass invariants by g2^3-27g3^2.  In the section "Elliptic
Modular Functions", the same equality is stated.  But it does not work,
unless d is given the factor (2w)^12.

So can anyone can explain the situation, why it works for J[z] but not
for n[z]? (I have a nagging feeling I'm using a wrong assumption
somewhere...)

-Titus


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