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23 Weierstrass Elliptic and Modular Functions Modular Functions 23.18 Modular Transformations 23.20 Mathematical Applications

§23.19 Interrelations

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Keywords:: interrelations, modular functions
Notes:: See Apostol (1990, Chapters 2, 3), Serre (1973, Chapter 7). (23.19.4) follows from (20.5.3) and (23.17.8).
Referenced by:: §23.15(i), §27.14(iv)
Permalink:: http://dlmf.nist.gov/23.19
See also:: Annotations for Ch.23

23.19.1 $\lambda\left(\tau\right)=16\left(\frac{{\eta}^{2}\left(2\tau\right)\eta\left(% \tfrac{1}{2}\tau\right)}{{\eta}^{3}\left(\tau\right)}\right)^{8},$

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Symbols:: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §23.19 and Ch.23

23.19.2 $J\left(\tau\right)=\frac{4}{27}\frac{\left(1-\lambda\left(\tau\right)+{\lambda% }^{2}\left(\tau\right)\right)^{3}}{\left(\lambda\left(\tau\right)\left(1-% \lambda\left(\tau\right)\right)\right)^{2}},$

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Symbols:: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant, $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §23.19 and Ch.23

23.19.3 $J\left(\tau\right)=\frac{{g_{2}}^{3}}{{g_{2}}^{3}-27{g_{3}}^{2}},$

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Symbols:: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §23.19 and Ch.23

where $g_{2},g_{3}$ are the invariants of the lattice $\mathbb{L}$ with generators 1 and $\tau$ ; see §23.3(i).

Also, with $\Delta$ defined as in (23.3.4),

23.19.4 $\Delta=(2\pi)^{12}{\eta}^{24}\left(\tau\right).$

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Symbols:: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\Delta$ : discriminant and $\tau$ : complex variable
Referenced by:: §23.19
Permalink:: http://dlmf.nist.gov/23.19.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §23.19 and Ch.23

23.18 Modular Transformations 23.20 Mathematical Applications

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