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

```

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