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Re: Typo in DedekindEta function definition?

  • To: mathgroup at
  • Subject: [mg71230] Re: Typo in DedekindEta function definition?
  • From: "sashap" <pavlyk at>
  • Date: Sat, 11 Nov 2006 03:38:03 -0500 (EST)
  • References: <ej1ops$dlq$>

Hi Titus,

There is no typo, just an implicit assumption that omega is normalized
to 1.

To understand what is going on look at

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 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

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