Re: Typo in DedekindEta function definition?

*To*: mathgroup at smc.vnet.net*Subject*: [mg71230] Re: Typo in DedekindEta function definition?*From*: "sashap" <pavlyk at gmail.com>*Date*: Sat, 11 Nov 2006 03:38:03 -0500 (EST)*References*: <ej1ops$dlq$1@smc.vnet.net>

Hi Titus, There is no typo, just an implicit assumption that omega is normalized to 1. See http://functions.wolfram.com/EllipticFunctions/DedekindEta/27/01/01/ To understand what is going on look at http://functions.wolfram.com/EllipticFunctions/WeierstrassInvariants/02/ You see that invariants would scale if both omega and omega prime are multiplied with the same number. KleinInvariantJ is scale invariant, but the modular discriminant g2^3-27*g3^2 is not. Oleksandr Pavlyk Special Functions Developer titus_piezas at yahoo.com wrote: > 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