Klein%20complete%20invariant

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1: 23.19 Interrelations

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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:

$J\left(\tau \right)$: Klein’s complete invariant, $\lambda \left(\tau \right)$: elliptic modular function and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

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23.19.3 $$J\left(\tau \right)=\frac{g_2^3}{g_2^3-27g_3^2},$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $𝕃$: lattice and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

►where $g_2,g_3$ are the invariants of the lattice $𝕃$ with generators $1$ and $\tau$; see §23.3(i). …

2: 23.16 Graphics

… ►See Figures 23.16.1–23.16.3 for the modular functions $\lambda$, $J$, and $\eta$. … ►

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Figure 23.16.1: Modular functions $\lambda \left(\mathrm{i}y\right)$, $J\left(\mathrm{i}y\right)$, $\eta \left(\mathrm{i}y\right)$ for $0\le y\le 3$. … Magnify

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3: 23.17 Elementary Properties

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$J\left(\mathrm{i}\right)=1,$

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$J\left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)=0,$

… ►For further results for $J\left(\tau \right)$ see Cohen (1993, p. 376). … ►

23.17.5 $$1728J\left(\tau \right)=q^{-2}+744+\mathrm{1\hspace{0.33em}96884}{q}^2+\mathrm{214\hspace{0.33em}93760}{q}^4+\mathrm{\cdots },$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $q$: nome and $\tau$: complex variable

Referenced by:

§23.17(ii)

Permalink:

http://dlmf.nist.gov/23.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

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4: 23.18 Modular Transformations

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Klein’s Complete Invariant

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23.18.4 $$J\left(𝒜\tau \right)=J\left(\tau \right).$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $\tau$: complex variable and $𝒜$: bilinear transformation

Referenced by:

§23.18

Permalink:

http://dlmf.nist.gov/23.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

► $J\left(\tau \right)$ is a modular form of level zero for SL$(2,\mathbb{Z})$. …

5: 23.15 Definitions

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23.15.1 $$q=\exp \left(-\pi \frac{K^{\prime }\left(k\right)}{K\left(k\right)}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function, $q$: nome and $k$: modulus

Permalink:

http://dlmf.nist.gov/23.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

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§23.15(ii) Functions $\lambda \left(\tau \right)$, $J\left(\tau \right)$, $\eta \left(\tau \right)$

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Klein’s Complete Invariant

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23.15.7 $$J\left(\tau \right)=\frac{{\left({\theta }_2^8(0,q)+{\theta }_3^8(0,q)+{\theta }_4^8(0,q)\right)}^3}{54{\left({\theta }_1^{\prime }(0,q)\right)}^8},$$

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Defines:

$J\left(\tau \right)$: Klein’s complete invariant

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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6: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
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$K\left(k\right)$, $K^{\prime }\left(k\right)$	complete elliptic integrals (§19.2(i)).
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$\mathrm{\Delta }$	discriminant $g_2^3-27g_3^2$.
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►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

7: 20 Theta Functions

Chapter 20 Theta Functions

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8: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

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KLEIN: Coulomb functions for real $\lambda$ and positive energy to high accuracy. Comput. Phys. Comm. 24 (2), pp. 141–159.

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Notes:

Double-precision Fortran code for positive energies.

External Links:

Document, Code

Referenced by:

5th item, §33.23(vii), 1st item.

Encodings:

BibTeX

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

BibTeX

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From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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External Links:

ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

BibTeX

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9: 15.17 Mathematical Applications

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§15.17(ii) Conformal Mappings

… ►See Klein (1894) and Hochstadt (1971). Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. …

10: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …