sigma function

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21: 27.11 Asymptotic Formulas: Partial Sums

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27.11.4 $$\sum _{n\le x}{\sigma }_1\left(n\right)=\frac{{\pi }^2}{12}{x}^2+O\left(x\ln x\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III (in slightly different form)

Permalink:

http://dlmf.nist.gov/27.11.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.5 $$\sum _{n\le x}{\sigma }_{\alpha }\left(n\right)=\frac{\zeta \left(\alpha +1\right)}{\alpha +1}{x}^{\alpha +1}+O\left(x^{\beta }\right),$$ $\alpha >0$, $\alpha \ne 1$, $\beta =\max (1,\alpha )$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

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22: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

Encodings:

BibTeX

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23: 19.2 Definitions

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19.2.2 $$r(s,t)=\frac{(p_1+p_{2}s)(p_3-p_{4}s)s}{(p_3+p_{4}s)(p_3-p_{4}s)s}=\frac{\rho }{s}+\sigma ,$$

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Symbols:

$r(s,t)$: rational function, $\rho (t)$: rational function, $\sigma (t)$: rational function and $p_j$: polynomial

Permalink:

http://dlmf.nist.gov/19.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(i), §19.2 and Ch.19

►where $p_j$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. …

24: 27.14 Unrestricted Partitions

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27.14.7 $$np\left(n\right)=\sum _{k=1}^n{\sigma }_1\left(k\right)p\left(n-k\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer and $n$: positive integer

A&S Ref:

24.2.1 II.A (in slightly different form) 24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20, Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/27.14.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

The erroneous ${\sigma }_1\left(n\right)$ in
$np\left(n\right)={\sum }_{k=1}^n{\sigma }_1\left(n\right)p\left(n-k\right)$
was replaced.

Reported 2015-08-17 by Howard Cohl and Hannah Cohen

See also:

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

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27.14.20 $$\tau \left(n\right)\equiv {\sigma }_{11}\left(n\right)(mod691).$$

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Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\equiv$: modular equivalence and $n$: positive integer

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E20

Encodings:

pMML, png, TeX

See also:

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

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25: 28.8 Asymptotic Expansions for Large $q$

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§28.8(ii) Sips’ Expansions

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§28.8(iii) Goldstein’s Expansions

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Barrett’s Expansions

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►

26: 19.25 Relations to Other Functions

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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

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

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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

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27: 30.15 Signal Analysis

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30.15.3 $${\int }_{-\tau }^{\tau }\frac{\sin \sigma (t-s)}{\pi (t-s)}{\varphi }_n(s)ds={\mathrm{\Lambda }}_n{\varphi }_n(t).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $n\ge m$: integer degree, $\tau >0$: parameter, $\sigma >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$

Permalink:

http://dlmf.nist.gov/30.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(ii), §30.15 and Ch.30

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30.15.4 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}t\omega }{\varphi }_n(t)dt={(-\mathrm{i})}^n\sqrt{\frac{2\pi \tau }{\sigma {\mathrm{\Lambda }}_n}}{\varphi }_n\left(\frac{\tau }{\sigma }\omega \right){\chi }_{\sigma }(\omega ),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, $\sigma >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$

Referenced by:

§30.15(iii)

Permalink:

http://dlmf.nist.gov/30.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(iii), §30.15 and Ch.30

… ►Equations (30.15.4) and (30.15.6) show that the functions ${\varphi }_n$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma ,\sigma ]$. … ►The sequence ${\varphi }_n$, $n=0,1,2,\mathrm{\dots }$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^2(-\tau ,\tau )$. … ►

30.15.9 $$\beta =\frac{1}{2\pi }{\int }_{-\sigma }^{\sigma }{\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}t\omega }f(t)dt\right|}^{2}d\omega$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sigma >0$: parameter and $f$: function

Permalink:

http://dlmf.nist.gov/30.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(v), §30.15 and Ch.30

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28: 19.28 Integrals of Elliptic Integrals

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19.28.1 $${\int }_0^1{t}^{\sigma -1}{R}_F(0,t,1)dt=\frac{1}{2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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19.28.2 $${\int }_0^1{t}^{\sigma -1}{R}_G(0,t,1)dt=\frac{\sigma }{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Permalink:

http://dlmf.nist.gov/19.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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19.28.3 $${\int }_0^1{t}^{\sigma -1}(1-t)R_D(0,t,1)dt=\frac{3}{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2.$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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19.28.4 $${\int }_0^1{t}^{\sigma -1}{(1-t)}^{c-1}{R}_{-a}(b_1,b_2;t,1)dt=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(\sigma \right)\mathrm{\Gamma }\left(\sigma +b_2-a\right)}{\mathrm{\Gamma }\left(\sigma +c-a\right)\mathrm{\Gamma }\left(\sigma +b_2\right)},$$ $c=b_1+b_2>0$, $\mathrm{\Re }\sigma >\max (0,a-b_2)$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sigma$: parameter

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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29: 32.11 Asymptotic Approximations for Real Variables

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32.11.19 $$w(x)=\sigma \sqrt{\frac{1}{2}x}+\sigma \rho {(2x)}^{-1/4}\cos \left(\psi (x)+\theta \right)+O\left(x^{-1}\right),$$ $x\to +\mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\cos z$: cosine function, $x$: real, $\sigma$: real constant, $\rho$: real constant, $\theta$: real constant and $\psi (z)$: function

Permalink:

http://dlmf.nist.gov/32.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iii), §32.11 and Ch.32

►where $\sigma$, $\rho$ $(>0)$, and $\theta$ are real constants, and … ►

32.11.27 $$\sigma =(2/\pi )\arcsin \left(\pi \lambda \right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arcsin z$: arcsine function and $\sigma$: constant

Permalink:

http://dlmf.nist.gov/32.11.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iv), §32.11 and Ch.32

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32.11.28 $$B=2^{-2\sigma }\frac{{\mathrm{\Gamma }}^2\left(\frac{1}{2}(1-\sigma )\right)\mathrm{\Gamma }\left(\frac{1}{2}(1+\sigma )+\nu \right)}{{\mathrm{\Gamma }}^2\left(\frac{1}{2}(1+\sigma )\right)\mathrm{\Gamma }\left(\frac{1}{2}(1-\sigma )+\nu \right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\nu$: parameter, $B$: constant and $\sigma$: constant

Permalink:

http://dlmf.nist.gov/32.11.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iv), §32.11 and Ch.32

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30: 2.10 Sums and Sequences

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On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

2.10.30 $$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$ $z\to z_j$,

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

where ${\sigma }_j$ is a positive constant.

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2.10.32 $$f^{(m)}(z)-g^{(m)}(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

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