interrelation between bases

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21: 19.25 Relations to Other Functions

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19.25.28 $$\mathrm{\Delta }(\mathrm{p},\mathrm{q})={\mathrm{p}\mathrm{s}}^2(u,k)-{\mathrm{q}\mathrm{s}}^2(u,k)=-\mathrm{\Delta }(\mathrm{q},\mathrm{p}),$$

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Defines:

$\mathrm{\Delta }(\mathrm{p},\mathrm{q})$: function (locally)

Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.25.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(v), §19.25 and Ch.19

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22: 13.14 Definitions and Basic Properties

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13.14.3 $$W_{\kappa ,\mu }\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}+\mu }U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z),$$

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Defines:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.1.33

Referenced by:

§13.14(i), §13.16(ii), §13.19, §13.20(i), §13.22, §13.4(iii), §18.34(i), §18.39(i)

Permalink:

http://dlmf.nist.gov/13.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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13.14.4 $$M(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}{z}^{-\frac{1}{2}b}{M}_{\frac{1}{2}b-a,\frac{1}{2}b-\frac{1}{2}}\left(z\right),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§10.16, (13.11.2), §13.11, §13.14(iii), §13.14(vii), §13.16(i), §13.18(i), §13.18(ii), §13.18(iii), §13.18(iv), §13.18(v), §13.23(iii), §13.25, §13.8(ii), §13.8(ii), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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13.14.5 $$U(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}{z}^{-\frac{1}{2}b}{W}_{\frac{1}{2}b-a,\frac{1}{2}b-\frac{1}{2}}\left(z\right).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§10.16, §13.11, §13.14(iii), §13.14(vii), §13.16(i), §13.18(i), §13.18(ii), §13.18(iii), §13.18(iv), §13.18(v), §13.23(iii), §13.8(ii), §13.8(ii), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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13.14.10 $$M_{\kappa ,\mu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}}{M}_{-\kappa ,\mu }\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: 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 and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(ii), §13.14 and Ch.13

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§13.14(vii) Connection Formulas

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23: 4.16 Elementary Properties

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Table 4.16.3: Trigonometric functions: interrelations. …

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	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
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24: Simon Ruijsenaars

… ►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …

25: 5.20 Physical Applications

… ►In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift $\mathrm{ph}\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)$; see (33.2.10) and Clark (1979). … ►

5.20.3 $${\psi }_n(\beta )={\int }_{{ℝ}^n}{\mathrm{e}}^{-\beta W}dx={(2\pi )}^{n/2}{\beta }^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^n\mathrm{\Gamma }\left(1+\frac{1}{2}j\beta \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $ℝ$: real line, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

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5.20.4

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $j$: nonnegative integer, $n$: nonnegative integer and $W$: potential energy

Permalink:

http://dlmf.nist.gov/5.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

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5.20.5 $${\psi }_n(\beta )=\frac{1}{{(2\pi )}^n}{\int }_{{[-\pi ,\pi ]}^n}{\mathrm{e}}^{-\beta W}d{\theta }_1\mathrm{\cdots }d{\theta }_n=\mathrm{\Gamma }\left(1+\frac{1}{2}n\beta \right){(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right))}^{-n}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $n$: nonnegative integer, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

… ►Veneziano (1968) identifies relationships between particle scattering amplitudes described by the beta function, an important early development in string theory. …

26: Guide to Searching the DLMF

… ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. … ►Sometimes there are distinctions between various special function names based on font style, such as the use of bold or calligraphic letters. …

27: 10.47 Definitions and Basic Properties

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§10.47(iv) Interrelations

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28: 18.26 Wilson Class: Continued

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§18.26(ii) Limit Relations

… ►See also Figure 18.21.1. …

29: Bille C. Carlson

… ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …

30: Bibliography R

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Elliptische Functionen. Teubner, Leipzig.

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Notes:

Based on notes of lectures by B. Riemann edited and published by H. Stahl. Reprinted by UMI Books on Demand (Ann Arbor, MI, USA).

External Links:

ZentralBlatt

Referenced by:

§20.12(ii).

Encodings:

BibTeX

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