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1: 1.16 Distributions

… ►A mapping $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is a linear functional if …$\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►A tempered distribution is a continuous linear functional $\mathrm{\Lambda }$ on $𝒯$. … ►A distribution in ${ℝ}^n$ is a continuous linear functional on ${𝒟}_n$. … ►Tempered distributions are continuous linear functionals on this space of test functions. …

2: 15.19 Methods of Computation

… ►For $z\in ℝ$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$. … ►This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. … ► …

3: 35.11 Tables

… ►Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

4: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

5: 21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

6: 2.6 Distributional Methods

… ►, a continuous linear functional) on the space $𝒯$ of rapidly decreasing functions on $ℝ$. …

7: Bibliography G

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The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:

ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

Encodings:

BibTeX

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8: 7.21 Physical Applications

… ►Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …

9: 15.8 Transformations of Variable

… ►

§15.8(i) Linear Transformations

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15.8.12 $$𝐅(a,b;a+b-m;z)={(1-z)}^{-m}𝐅(\tilde{a},\tilde{b};\tilde{a}+\tilde{b}+m;z),$$ $\tilde{a}=a-m,\tilde{b}=b-m$.

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/15.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

… ►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. … ►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). …

10: Bibliography D

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Complex zeros of linear combinations of spherical Bessel functions and their derivatives. SIAM J. Math. Anal. 4 (1), pp. 128–133.

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External Links:

ISSN 1095-7154, Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§10.58.

Encodings:

BibTeX

… ►

Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:

With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)

External Links:

ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.

Encodings:

BibTeX

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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

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External Links:

ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

Encodings:

BibTeX

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

Encodings:

BibTeX

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Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.

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External Links:

ISSN 0022-2526, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§14.15(iii), §2.8(i).

Encodings:

BibTeX

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