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1: 23.20 Mathematical Applications

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§23.20(iv) Modular and Quintic Equations

►The modular equation of degree $p$, $p$ prime, is an algebraic equation in $\alpha =\lambda \left(p\tau \right)$ and $\beta =\lambda \left(\tau \right)$. For $p=2,3,5,7$ and with $u={\alpha }^{1/4}$, $v={\beta }^{1/4}$, the modular equation is as follows: … ►

23.20.8 $$(1-u^8)(1-v^8)={(1-uv)}^8,$$ $p=7$.

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Symbols:

$u$ , $v$ and $p$: degree

Permalink:

http://dlmf.nist.gov/23.20.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.20(iv), §23.20 and Ch.23

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2: Bibliography

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

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3: 27.14 Unrestricted Partitions

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§27.14(iv) Relation to Modular Functions

… ► $\eta \left(\tau \right)$ satisfies the following functional equation: if $a,b,c,d$ are integers with $ad-bc=1$ and $c>0$, then …

4: 21.5 Modular Transformations

… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. …

5: Bibliography B

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

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6: Ranjan Roy

… ►Roy has published many papers on differential equations, fluid mechanics, special functions, Fuchsian groups, and the history of mathematics. …He also authored another two advanced mathematics books: Sources in the development of mathematics (Roy, 2011), Elliptic and modular functions from Gauss to Dedekind to Hecke (Roy, 2017). …

7: 23.2 Definitions and Periodic Properties

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23.2.4 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{w\in 𝕃\setminus \{0\}}\left(\frac{1}{{(z-w)}^2}-\frac{1}{w^2}\right),$$

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Defines:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function

Symbols:

$\mathrm{\wp }(z;g_2,g_3)$: Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modular functions, Weierstrass $\mathrm{\wp }$-function

Referenced by:

§23.9, Erratum (V1.0.5) for Equation (23.2.4)

Permalink:

http://dlmf.nist.gov/23.2.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the denominator ${(z-w)}^2$ was given incorrectly as $(z-w^2)$.

Reported 2012-02-16 by James D. Walker

See also:

Annotations for §23.2(ii), §23.2 and Ch.23

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8: 17.2 Calculus

… ►For properties of the function $f\left(q\right)=q^{-1/24}\eta \left(\frac{\ln q}{2\pi \mathrm{i}}\right)={(q;q)}_{\mathrm{\infty }}$ see §27.14. … ►

17.2.6_1 $${(q;q)}_{\mathrm{\infty }}=\sqrt{\frac{2\pi }{t}}\exp \left(-\frac{{\pi }^2}{6t}+\frac{t}{24}\right){(\widehat{q};\widehat{q})}_{\mathrm{\infty }},$$ $\mathrm{\Re }t>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\mathrm{\Re }$: real part and $q$: complex base

Referenced by:

§17.2(i), §17.2(i), Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/17.2.E6_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.5):

This equation was added.

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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17.2.6_2 $${(-q;q)}_{\mathrm{\infty }}=\frac{1}{\sqrt{2}}\exp \left(\frac{{\pi }^2}{12t}+\frac{t}{24}\right){({\widehat{q}}^{\frac{1}{2}};\widehat{q})}_{\mathrm{\infty }},$$ $t>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

§17.2(i), Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/17.2.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.5):

This equation was added.

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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17.2.22 $$\frac{{(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n}{{(a^{\frac{1}{2}},-a^{\frac{1}{2}};q)}_n}=\frac{{(a{q}^2;q^2)}_n}{{(a;q^2)}_n}=\frac{1-a{q}^{2n}}{1-a},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base and $n$: nonnegative integer

Referenced by:

Erratum (V1.0.16) for Equations (17.2.22) and (17.2.23)

Permalink:

http://dlmf.nist.gov/17.2.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The numerator of the leftmost fraction was corrected to read ${(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n$ instead of ${(q{a}^{\frac{1}{2}},-a{q}^{\frac{1}{2}};q)}_n$.

Reported 2017-06-26 by Jason Zhao

See also:

Annotations for §17.2(i), §17.2 and Ch.17

… ► $q$-differential equations are considered in §17.6(iv). …

9: 23.18 Modular Transformations

§23.18 Modular Transformations

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Elliptic Modular Function

► $\lambda \left(𝒜\tau \right)$ equals … ► $J\left(\tau \right)$ is a modular form of level zero for SL$(2,\mathbb{Z})$. … ►Note that $\eta \left(\tau \right)$ is of level $\frac{1}{2}$. …

10: 23.21 Physical Applications

§23.21 Physical Applications

… ► … ►

§23.21(iv) Modular Functions

►Physical applications of modular functions include: … ►

String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).