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11: 20.9 Relations to Other Functions

… ►

§20.9(ii) Elliptic Functions and Modular Functions

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). ►As a function of $\tau$, $k^2$ is the elliptic modular function; see Walker (1996, Chapter 7) and (23.15.2), (23.15.6). …

12: Ranjan Roy

… ►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). …

13: 27.14 Unrestricted Partitions

… ►

§27.14(iv) Relation to Modular Functions

►Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by ►

27.14.12 $$\eta \left(\tau \right)={\mathrm{e}}^{\pi \mathrm{i}\tau /12}\prod _{n=1}^{\mathrm{\infty }}(1-{\mathrm{e}}^{2\pi \mathrm{i}n\tau }),$$ $\mathrm{\Im }\tau >0$.

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Defines:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $n$: positive integer

Referenced by:

§23.15(ii), §27.14(vi)

Permalink:

http://dlmf.nist.gov/27.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iv), §27.14 and Ch.27

… ►For further properties of the function $\eta \left(\tau \right)$ see §§23.15–23.19. … ►

27.14.16 $$\mathrm{\Delta }\left(\tau \right)={(2\pi )}^{12}{(\eta \left(\tau \right))}^{24},$$ $\mathrm{\Im }\tau >0$,

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Defines:

$\mathrm{\Delta }\left(\tau \right)$: discriminant function

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter and $\mathrm{\Im }$: imaginary part

Permalink:

http://dlmf.nist.gov/27.14.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

…

14: 21.5 Modular Transformations

… ►

§21.5(i) Riemann Theta Functions

… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi (𝚪)$ is determinate: … ►

§21.5(ii) Riemann Theta Functions with Characteristics

… ►For explicit results in the case $g=1$, see §20.7(viii).

15: William P. Reinhardt

… ►

23 Weierstrass Elliptic and Modular Functions

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16: 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:

modularfunctions, 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

►

23.2.5 $$\zeta \left(z\right)=\frac{1}{z}+\sum _{w\in 𝕃\setminus \{0\}}\left(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right),$$

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Defines:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function

Symbols:

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

Keywords:

modularfunctions, Weierstrass zeta function

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.2.E5

Encodings:

pMML, png, TeX

See also:

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

►

23.2.6 $$\sigma \left(z\right)=z\prod _{w\in 𝕃\setminus \{0\}}\left(\left(1-\frac{z}{w}\right)\exp \left(\frac{z}{w}+\frac{z^2}{2w^2}\right)\right).$$

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Defines:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function

Symbols:

$g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\sigma (z;g_2,g_3)$: Weierstrass sigma function, $\in$: element of, $\exp z$: exponential function, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modularfunctions, Weierstrass sigma function

Referenced by:

§23.2(iii)

Permalink:

http://dlmf.nist.gov/23.2.E6

Encodings:

pMML, png, TeX

See also:

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

…

17: 23.22 Methods of Computation

… ►The modular functions $\lambda \left(\tau \right)$, $J\left(\tau \right)$, and $\eta \left(\tau \right)$ are also obtainable in a similar manner from their definitions in §23.15(ii). …

18: 20.3 Graphics

… ►

►►

Figure 20.3.2: ${\theta }_1(\pi x,q)$, $0\le x\le 2$, $q$ = 0. …Here $q^{\text{Dedekind}}={\mathrm{e}}^{-\pi y_0}=0.19$ approximately, where $y=y_0$ corresponds to the maximum value of Dedekind’s eta function $\eta \left(\mathrm{i}y\right)$ as depicted in Figure 23.16.1. Magnify

…

19: Bibliography R

… ►

Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

…

20: 20.7 Identities

… ►

20.7.33 $${(-\mathrm{i}\tau )}^{1/2}{\theta }_4\left(z\right|\tau )=\exp \left(\mathrm{i}{\tau }^{\prime }{z}^2/\pi \right){\theta }_2\left(z{\tau }^{\prime }\right|{\tau }^{\prime }).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter

Referenced by:

(20.10.3), §20.13

Permalink:

http://dlmf.nist.gov/20.7.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(viii), §20.7 and Ch.20

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