of integers

SearchAdvancedHelp

(0.004 seconds)

31—40 of 562 matching pages

31: 24.1 Special Notation

… ► ►►

$j,k,\mathrm{\ell },m,n$	integers, nonnegative unless stated otherwise.
…

►Unless otherwise noted, the formulas in this chapter hold for all values of the variables $x$ and $t$, and for all nonnegative integers $n$. …

32: 27.1 Special Notation

… ► ►►►

$d,k,m,n$	positive integers (unless otherwise indicated).
…
$p,p_1,p_2,\mathrm{\dots }$	prime numbers (or primes): integers ($>1$) with only two positive integer divisors, $1$ and the number itself.
…

33: 21.1 Special Notation

… ► ►►►►

$g,h$	positive integers.
${\mathbb{Z}}^g$	$\mathbb{Z}\times \mathbb{Z}\times \mathrm{\cdots }\times \mathbb{Z}$ ($g$ times).
…
${\mathbb{Z}}^{g\times h}$	set of all $g\times h$ matrices with integer elements.
…

…

34: 27.5 Inversion Formulas

… ►

27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

ⓘ

Symbols:

$d$: positive integer, $n$: positive integer, $f$: function, $g$: function and $h$: function

Permalink:

http://dlmf.nist.gov/27.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

… ►

27.5.2 $$\sum _{d|n}\mu \left(d\right)=\lfloor \frac{1}{n}\rfloor ,$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $\lfloor x\rfloor$: floor of $x$ , $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.1 II.B (in slightly different form)

Referenced by:

§27.5

Permalink:

http://dlmf.nist.gov/27.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

… ►

27.5.3 $$g(n)=\sum _{d|n}f(d)⟺f(n)=\sum _{d|n}g(d)\mu \left(\frac{n}{d}\right).$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

… ►

27.5.4 $$n=\sum _{d|n}\varphi \left(d\right)⟺\varphi \left(n\right)=\sum _{d|n}d\mu \left(\frac{n}{d}\right),$$

ⓘ

Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.2 II.B

Permalink:

http://dlmf.nist.gov/27.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

… ►

27.5.8 $$g(n)=\prod _{d|n}f(d)⟺f(n)=\prod _{d|n}{\left(g\left(\frac{n}{d}\right)\right)}^{\mu \left(d\right)}.$$

ⓘ

Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

…

35: 27.8 Dirichlet Characters

… ►If $k$ $(>1)$ is a given integer, then a function $\chi \left(n\right)$ is called a Dirichlet character (mod $k$) if it is completely multiplicative, periodic with period $k$, and vanishes when $\left(n,k\right)>1$. … ►

27.8.2 $$\chi \left(mn\right)=\chi \left(m\right)\chi \left(n\right),$$ $m,n=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

►

27.8.3 $$\chi \left(n+k\right)=\chi \left(n\right),$$ $n=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

►

27.8.4 $$\chi \left(n\right)=0,$$ $\left(n,k\right)>1$.

ⓘ

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.8 and Ch.27

… ►A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. …

36: 4.8 Identities

… ►

4.8.5 $$\mathrm{Ln}\left(z^n\right)=n\mathrm{Ln}z,$$ $n\in \mathbb{Z}$,

ⓘ

Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $n$: integer and $z$: complex variable

A&S Ref:

4.1.10

Referenced by:

§4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

►

4.8.6 $$\ln \left(z^n\right)=n\ln z,$$ $n\in \mathbb{Z}$, $-\pi \le n\mathrm{ph}z\le \pi$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathbb{Z}$: set of all integers, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $n$: integer and $z$: complex variable

A&S Ref:

4.1.11

Permalink:

http://dlmf.nist.gov/4.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.8 $$\mathrm{Ln}\left(\exp z\right)=z+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer and $z$: complex variable

A&S Ref:

4.2.2

Permalink:

http://dlmf.nist.gov/4.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►where the integer $k$ is chosen so that $\mathrm{\Re }\left(-\mathrm{i}z\ln a\right)+2k\pi \in [-\pi ,\pi ]$. … ►The restriction on $z_1$ can be removed when $z_2$ is an integer.

37: 14.34 Software

… ►

Adams and Swarztrauber (1997). Integer parameters. Fortran.

►

Braithwaite (1973). Integer parameters. Fortran.

►

Delic (1979a). Integer parameters. Fortran.

►

Gil and Segura (1997). Integer and half-integer parameters. Fortran.

►

Gil and Segura (1998). Integer parameters. Fortran.

…

38: 27.13 Functions

… ►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. The basic problem is that of expressing a given positive integer $n$ as a sum of integers from some prescribed set $S$ whose members are primes, squares, cubes, or other special integers. … ►Waring’s problem is to find, for each positive integer $k$, whether there is an integer $m$ (depending only on $k$) such that the equation …has nonnegative integer solutions for all $n\ge 1$. …Similarly, $G\left(k\right)$ denotes the smallest $m$ for which (27.13.1) has nonnegative integer solutions for all sufficiently large $n$. …

39: 28.4 Fourier Series

… ►

28.4.9 $$2{\left(A_0^{2n}(q)\right)}^2+\sum _{m=1}^{\mathrm{\infty }}{\left(A_{2m}^{2n}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

A&S Ref:

20.5.4

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

►

28.4.10 $$\sum _{m=0}^{\mathrm{\infty }}{\left(A_{2m+1}^{2n+1}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

►

28.4.11 $$\sum _{m=0}^{\mathrm{\infty }}{\left(B_{2m+1}^{2n+1}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

►

28.4.12 $$\sum _{m=0}^{\mathrm{\infty }}{\left(B_{2m+2}^{2n+2}(q)\right)}^2=1.$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $B_m(q)$: Fourier coefficient

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

… ►

28.4.17 $$A_{2m}^{2n}(-q)={(-1)}^{n-m}{A}_{2m}^{2n}(q),$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

A&S Ref:

20.8.5

Permalink:

http://dlmf.nist.gov/28.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(v), §28.4 and Ch.28

…

40: 29.14 Orthogonality

… ►

29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.6 $${\mathrm{𝑑𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑑𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.7 $${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(s,k^2){\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

… ►In each system $n$ ranges over all nonnegative integers and $m=0,1,\mathrm{\dots },n$. …