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Typo in DedekindEta function definition?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71194] Typo in DedekindEta function definition?
*From*: titus_piezas at yahoo.com
*Date*: Fri, 10 Nov 2006 06:37:53 -0500 (EST)
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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