reciprocity law

SearchAdvancedHelp

(0.002 seconds)

1—10 of 28 matching pages

1: 27.9 Quadratic Characters

… ►

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

ⓘ

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: 25.16 Mathematical Applications

… ►which satisfies the reciprocity law …

3: 36.6 Scaling Relations

§36.6 Scaling Relations

…

4: 5.3 Graphics

… ►

►►

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

… ►

►

►

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

…

5: Bibliography F

… ►

On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

ⓘ

External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

… ►

On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

ⓘ

External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

Encodings:

BibTeX

…

6: 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). …

7: 8.24 Physical Applications

… ►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …

8: 5.2 Definitions

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

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

…

9: 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 $t$. …

10: 27.14 Unrestricted Partitions

… ►Euler introduced the reciprocal of the infinite product ►

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

ⓘ

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:

pMML, png, TeX

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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:

pMML, png, TeX

See also:

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

…