of reciprocals

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1: 27.9 Quadratic Characters

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27.9.2 $$(2|p)={(-1)}^{(p^2-1)/8}.$$

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Symbols:

$(n|p)$: Legendre symbol and $p$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … ►Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$.

2: 5.3 Graphics

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Figure 5.3.1: $\mathrm{\Gamma }\left(x\right)$ and $1/\mathrm{\Gamma }\left(x\right)$. … Magnify

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Figure 5.3.5: $1/|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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3: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

4: 5.2 Definitions

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5.2.1 $$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $\mathrm{\Re }z>0$.

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Defines:

$\mathrm{\Gamma }\left(z\right)$: gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

6.1.1

Referenced by:

§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)

Permalink:

http://dlmf.nist.gov/5.2.E1

Encodings:

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See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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5: 27.16 Cryptography

… ►To do this, let $s$ denote the reciprocal of $r$ modulo $\varphi \left(n\right)$, so that $rs=1+t\varphi \left(n\right)$ for some integer $t$. …

6: 27.14 Unrestricted Partitions

… ►Euler introduced the reciprocal of the infinite product ►

27.14.2 $$f\left(x\right)=\prod _{m=1}^{\mathrm{\infty }}(1-x^m)={(x;x)}_{\mathrm{\infty }},$$ ,

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Defines:

$f\left(x\right)$: Euler’s reciprocal function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $m$: positive integer and $x$: real number

Referenced by:

§27.14(iv), §27.21, Erratum (V1.0.28) for Equation (27.14.2)

Permalink:

http://dlmf.nist.gov/27.14.E2

Encodings:

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Addition (effective with 1.0.28):

The representation in terms of ${(x;x)}_{\mathrm{\infty }}$ was added to this equation.

See also:

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

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27.14.3 $$\frac{1}{f\left(x\right)}=\sum _{n=0}^{\mathrm{\infty }}p\left(n\right)x^n,$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

A&S Ref:

24.2.1 I.B

Permalink:

http://dlmf.nist.gov/27.14.E3

Encodings:

pMML, png, TeX

See also:

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

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27.14.4 $$f\left(x\right)=1-x-x^2+x^5+x^7-x^{12}-x^{15}+\mathrm{\cdots }=1+\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k\left(x^{\omega (k)}+x^{\omega (-k)}\right),$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $k$: positive integer, $x$: real number and $\omega (k)$: pentagonal numbers

Permalink:

http://dlmf.nist.gov/27.14.E4

Encodings:

pMML, png, TeX

See also:

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

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27.14.15 $$5\frac{{(f\left(x^5\right))}^5}{{(f\left(x\right))}^6}=\sum _{n=0}^{\mathrm{\infty }}p\left(5n+4\right)x^n$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.14.E15

Encodings:

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See also:

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

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7: 5.7 Series Expansions

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§5.7(i) Maclaurin and Taylor Series

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8: 5.8 Infinite Products

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5.8.3 $${\left|\frac{\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x+\mathrm{i}y\right)}\right|}^2=\prod _{k=0}^{\mathrm{\infty }}\left(1+\frac{y^2}{{(x+k)}^2}\right),$$ $x\ne 0,-1,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable

A&S Ref:

6.1.25 (where the formula for the reciprocal is given.)

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E3

Encodings:

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See also:

Annotations for §5.8 and Ch.5

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9: 5.19 Mathematical Applications

… ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …

10: 5.23 Approximations

… ►Clenshaw (1962) also gives 20D Chebyshev-series coefficients for $\mathrm{\Gamma }\left(1+x\right)$ and its reciprocal for $0\le x\le 1$. …