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