sigma function

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1: 32.6 Hamiltonian Structure

… ►The function … ►

32.6.7 $$q=-{\sigma }^{\prime },$$

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Symbols:

$q$: coordinate and $\sigma$: function

Permalink:

http://dlmf.nist.gov/32.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

►

32.6.8 $$p=-{\sigma }^{\prime \prime },$$

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Symbols:

$p$: momentum and $\sigma$: function

Permalink:

http://dlmf.nist.gov/32.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

… ►The function $\sigma (z)={\mathrm{H}}_{\text{II}}(q,p,z)$ defined by (32.6.9) satisfies … ►The function …

2: 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 $\eta \left(\tau \right)$. …

3: 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, Weierstrass sigmafunction

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).

4: 10.73 Physical Applications

… ►For applications of the Rayleigh function ${\sigma }_n\left(\nu \right)$ (§10.21(xiii)) to problems of heat conduction and diffusion in liquids see Kapitsa (1951a).

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

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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, $n$: 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

… ►

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

…

6: 33.13 Complex Variable and Parameters

… ►

33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

…

7: 26.2 Basic Definitions

… ►The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. …

8: 10.21 Zeros

… ►If ${\sigma }_{\nu }$ is a zero of ${𝒞}_{\nu }^{\prime }\left(z\right)$, then … ►The parameter $t$ may be regarded as a continuous variable and ${\rho }_{\nu }$, ${\sigma }_{\nu }$ as functions ${\rho }_{\nu }(t)$, ${\sigma }_{\nu }(t)$ of $t$. … ►The functions ${\rho }_{\nu }(t)$ and ${\sigma }_{\nu }(t)$ are related to the inverses of the phase functions ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ defined in §10.18(i): if $\nu \ge 0$, then … ►The Rayleigh function ${\sigma }_n\left(\nu \right)$ is defined by ►

10.21.55 $${\sigma }_n\left(\nu \right)=\sum _{m=1}^{\mathrm{\infty }}{(j_{\nu ,m})}^{-2n},$$ $n=1,2,3,\mathrm{\dots }$.

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Defines:

${\sigma }_n\left(\nu \right)$: Rayleigh function

Symbols:

$m$: integer, $n$: integer and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.21.E55

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(xiii), §10.21 and Ch.10

…

9: 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 $\sigma \left(z\right)$. …

10: 23.6 Relations to Other Functions

… ►

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

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

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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, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: 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

… ►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). … ►For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma \left(z\right)$ and $\mathrm{\wp }\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta \left(z\right)$ see Lawden (1989, §8.11). …