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

§23.19 Interrelations

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

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

Symbols:: $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ : Dedekind’s eta function (or Dedekind modular function), $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ : elliptic modular function and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E1
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{J\/}\nolimits\!\left(\tau\right)$ : Klein’s complete invariant, $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ : elliptic modular function and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E2
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{J\/}\nolimits\!\left(\tau\right)$ : Klein’s complete invariant, $g_{2}$ , $g_{3}$ : lattice invariants and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E3
Encodings:: TeX, pMML, png

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 $\Delta=(2\pi)^{{12}}{\mathop{\eta\/}\nolimits^{{24}}}\!\left(\tau\right).$

Symbols:: $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ : Dedekind’s eta function (or Dedekind modular function), $\Delta$ : discriminant and $\tau$ : complex variable
Referenced by:: §23.19
Permalink:: http://dlmf.nist.gov/23.19.E4
Encodings:: TeX, pMML, png

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