sigma function

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11: 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. …

12: 27.2 Functions

… ►Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. … ►

Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…

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13: 23.12 Asymptotic Approximations

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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

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14: 23.9 Laurent and Other Power Series

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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

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15: 27.4 Euler Products and Dirichlet Series

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27.4.11 $$\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)n^{-s}=\zeta \left(s\right)\zeta \left(s-\alpha \right),$$ $\mathrm{\Re }s>\max (1,1+\mathrm{\Re }\alpha )$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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16: 23.8 Trigonometric Series and Products

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

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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

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

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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

17: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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§33.10(i) Large $\rho$

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§33.10(ii) Large Positive $\eta$

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${\sigma }_0\left(\eta \right)=\eta (\ln \eta -1)+\frac{1}{4}\pi +o\left(1\right),$

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§33.10(iii) Large Negative $\eta$

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${\sigma }_0\left(\eta \right)=\eta (\ln \left(-\eta \right)-1)-\frac{1}{4}\pi +o\left(1\right),$

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18: 33.2 Definitions and Basic Properties

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33.2.7 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={(\mp \mathrm{i})}^{\mathrm{\ell }}{\mathrm{e}}^{(\pi \eta /2)\pm \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{W}_{\mp \mathrm{i}\eta ,\mathrm{\ell }+\frac{1}{2}}\left(\mp 2\mathrm{i}\rho \right),$$

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Defines:

$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions

Symbols:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§13.18(vi), §33.23(vi)

Permalink:

http://dlmf.nist.gov/33.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

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33.2.9 $${\theta }_{\mathrm{\ell }}(\eta ,\rho )=\rho -\eta \ln \left(2\rho \right)-\frac{1}{2}\mathrm{\ell }\pi +{\sigma }_{\mathrm{\ell }}\left(\eta \right),$$

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Defines:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions

Symbols:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.5.5

Referenced by:

§33.10(i), §33.11

Permalink:

http://dlmf.nist.gov/33.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

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33.2.10 $${\sigma }_{\mathrm{\ell }}\left(\eta \right)=\mathrm{ph}\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right),$$

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Defines:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

A&S Ref:

14.5.6

Referenced by:

§33.10(ii), §33.10(iii), §33.13, §33.2(iii), §33.25, §5.20

Permalink:

http://dlmf.nist.gov/33.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

… ►Also, ${\mathrm{e}}^{\mp \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{H}_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are analytic functions of $\eta$ when . …

19: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all $n\le {10}^4$: (a) the canonical factorization of $n$ into powers of primes; (b) the Euler totient $\varphi \left(n\right)$; (c) the divisor function $d\left(n\right)$; (d) the sum $\sigma (n)$ of these divisors. …Table III lists all solutions $n\le {10}^4$ of the equation $d\left(n\right)=m$, and Table IV lists all solutions $n$ of the equation $\sigma (n)=m$ for all $m\le {10}^4$. …6 lists $\varphi \left(n\right),d\left(n\right)$, and $\sigma (n)$ for $n\le 1000$; Table 24. …

20: 13.7 Asymptotic Expansions for Large Argument

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13.7.6 $$C_n=1,\chi (n),\left(\chi (n)+\sigma {\nu }^{2}n\right){\nu }^n,$$

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Defines:

$C_n$: coefficient (locally)

Symbols:

$n$: nonnegative integer, $\sigma$ , $\nu$ and $\chi (n)$: function

Permalink:

http://dlmf.nist.gov/13.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(ii), §13.7 and Ch.13

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