Weierstrass sigma function

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1: 23.1 Special Notation

… ►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 $\eta \left(\tau \right)$. …

2: 23.2 Definitions and Periodic Properties

… ►

§23.2(ii) Weierstrass Elliptic Functions

… ►

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:

modular functions, Weierstrasssigmafunction

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

… ►The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. … ►For $j=1,2,3$, the function $\sigma \left(z\right)$ satisfies … ►For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).

3: 23.10 Addition Theorems and Other Identities

… ►

23.10.3 $$\frac{\sigma \left(u+v\right)\sigma \left(u-v\right)}{{\sigma }^2\left(u\right){\sigma }^2\left(v\right)}=\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(u\right),$$

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Symbols:

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

A&S Ref:

18.4.4

Referenced by:

§23.10(ii)

Permalink:

http://dlmf.nist.gov/23.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(i), §23.10 and Ch.23

… ►For further addition-type identities for the $\sigma$-function see Lawden (1989, §6.4). … ►

23.10.10 $$\sigma \left(2z\right)=-{\mathrm{\wp }}^{\prime }\left(z\right){\sigma }^4\left(z\right).$$

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Symbols:

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

A&S Ref:

18.4.8

Referenced by:

§23.10(ii)

Permalink:

http://dlmf.nist.gov/23.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(ii), §23.10 and Ch.23

… ►

23.10.14 $$A_n=n\prod _{j=0}^{n-1}\prod _{\begin{array}{c}\mathrm{\ell }=0\\ \mathrm{\ell }\ne j\end{array}}^{n-1}\frac{1}{\sigma \left((2j{\omega }_1+2\mathrm{\ell }{\omega }_3)/n\right)}.$$

ⓘ

Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $𝕃$: lattice, $n$: integer, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $A_n$: coefficients

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iii), §23.10 and Ch.23

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23.10.19 $$\sigma \left(cz|c𝕃\right)=c\sigma \left(z|𝕃\right).$$

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Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $𝕃$: lattice, $z$: complex and $c$: constant

A&S Ref:

18.2.4

Permalink:

http://dlmf.nist.gov/23.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iv), §23.10 and Ch.23

…

4: 23.6 Relations to Other Functions

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23.6.9 $$\sigma \left(z\right)=2{\omega }_1\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\frac{{\theta }_1(\pi z/(2{\omega }_1),q)}{\pi {\theta }_1^{\prime }(0,q)},$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

A&S Ref:

18.10.8

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

►

23.6.10 $$\sigma \left({\omega }_1\right)=2{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1{\omega }_1\right){\theta }_2(0,q)}{\pi {\theta }_1^{\prime }(0,q)},$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $𝕃$: lattice, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

… ►

23.6.15 $$\frac{\sigma \left(u+{\omega }_j\right)}{\sigma \left({\omega }_j\right)}=\exp \left({\eta }_{j}u+\frac{{\eta }_1{u}^2}{2{\omega }_1}\right)\frac{{\theta }_{j+1}(z,q)}{{\theta }_{j+1}(0,q)},$$ $j=1,2,3$.

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\exp z$: exponential function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, ${\eta }_j$: complex number and $u$: change of variable

Referenced by:

§23.6(i), Erratum (V1.1.1) for Equation (23.6.15)

Permalink:

http://dlmf.nist.gov/23.6.E15

Encodings:

pMML, png, TeX

Correction (effective with 1.1.1):

The factor $\exp \left({\eta }_{j}u+\frac{{\eta }_j{u}^2}{2{\omega }_1}\right)$ has been corrected to be $\exp \left({\eta }_{j}u+\frac{{\eta }_1{u}^2}{2{\omega }_1}\right)$.

Suggested 2021-02-10 by Jan Felipe van Diejen

See also:

Annotations for §23.6(i), §23.6 and Ch.23

►For further results for the $\sigma$-function see Lawden (1989, §6.2). … ►For representations of the Jacobi functions $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ as quotients of $\sigma$-functions see Lawden (1989, §§6.2, 6.3). …

5: 23.3 Differential Equations

… ►As functions of $g_2$ and $g_3$, $\mathrm{\wp }(z;g_2,g_3)$ and $\zeta (z;g_2,g_3)$ are meromorphic and $\sigma (z;g_2,g_3)$ is entire. …

6: 23.4 Graphics

… ►Line graphs of the Weierstrass functions $\mathrm{\wp }\left(x\right)$, $\zeta \left(x\right)$, and $\sigma \left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … ►Surfaces for the Weierstrass functions $\mathrm{\wp }\left(z\right)$, $\zeta \left(z\right)$, and $\sigma \left(z\right)$. …

7: 23.9 Laurent and Other Power Series

… ►

23.9.7 $$\sigma \left(z\right)=\sum _{m,n=0}^{\mathrm{\infty }}{a}_{m,n}{(10c_2)}^m{(56c_3)}^n\frac{z^{4m+6n+1}}{(4m+6n+1)!},$$

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Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $!$: factorial (as in $n!$), $𝕃$: lattice, $n$: integer, $m$: integer, $z$: complex, $c_n$ and $a_{m,n}$: coefficients

A&S Ref:

18.5.6

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

…

8: 23.12 Asymptotic Approximations

… ►

23.12.3 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\pi }^2{z}^2}{24{\omega }_1^2}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\left(1-\left(\frac{{\pi }^2{z}^2}{{\omega }_1^2}-4{\sin }^2\left(\frac{\pi z}{2{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Permalink:

http://dlmf.nist.gov/23.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

…

9: 23.8 Trigonometric Series and Products

… ►

23.8.6 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\prod _{n=1}^{\mathrm{\infty }}\frac{1-2q^{2n}\cos \left(\pi z/{\omega }_1\right)+q^{4n}}{{(1-q^{2n})}^2},$$

ⓘ

Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(iii), §23.8 and Ch.23

►

23.8.7 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\eta }_1{z}^2}{2{\omega }_1}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left(\pi (2n{\omega }_3+z)/(2{\omega }_1)\right)\sin \left(\pi (2n{\omega }_3-z)/(2{\omega }_1)\right)}{{\sin }^2\left(\pi n{\omega }_3/{\omega }_1\right)}.$$

ⓘ

Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(iii), §23.8 and Ch.23

10: 19.25 Relations to Other Functions

… ►

19.25.40 $$z+2\omega =\pm \sigma \left(z\right)R_F({\sigma }_1^2(z),{\sigma }_2^2(z),{\sigma }_3^2(z)),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $z$: complex number and ${\sigma }_j(z)$: function

Proof sketch:

Combine Erdélyi et al. (1953b, §§13.12(22), 13.13(22)), (19.25.35) and (19.16.1).

Referenced by:

§19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E40

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

… ►

19.25.41 $${\sigma }_j(z)=\exp \left(-{\eta }_{j}z\right)\sigma \left(z+{\omega }_j\right)/\sigma \left({\omega }_j\right),$$ $j=1,2,3,$

ⓘ

Defines:

${\sigma }_j(z)$: function (locally)

Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\exp z$: exponential function, $j$: $(=1,2,3)$ , $z$: complex number, ${\omega }_j$: nonzero complex number and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/19.25.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

…