generalized Mehler–Fock transformation

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31: 8.24 Physical Applications

§8.24 Physical Applications

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§8.24(iii) Generalized Exponential Integral

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

32: 19.15 Advantages of Symmetry

… ►Symmetry in $x,y,z$ of $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. … ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. …

33: 15.11 Riemann’s Differential Equation

§15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►

§15.11(ii) Transformation Formulas

… ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by … ►for arbitrary $\lambda$ and $\mu$.

34: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►More generally in (19.14.4), … ►

§19.14(ii) General Case

… ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. …

35: 4.44 Other Applications

§4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). ►For an application of the Lambert $W$-function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). …

36: 4.12 Generalized Logarithms and Exponentials

§4.12 Generalized Logarithms and Exponentials

►A generalized exponential function $\varphi (x)$ satisfies the equations …Its inverse $\psi (x)$ is called a generalized logarithm. It, too, is strictly increasing when $0\le x\le 1$, and … ►For analytic generalized logarithms, see Kneser (1950).

37: Bibliography W

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Caustics: Mathematical curves generated by light shined through rippled plastic. Scientific American 249, pp. 146–153.

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Referenced by:

§36.14(ii).

Encodings:

BibTeX

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

Encodings:

BibTeX

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Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

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On a generalization of the functions $\mathrm{ber}$ x, $\mathrm{bei}$ x, $\ker$ x, $\mathrm{kei}$ x. Quart. J. Pure Appl. Math. 42, pp. 316–342.

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External Links:

ZentralBlatt

Referenced by:

§10.61(iii), §10.61(iv), §10.65(i), §10.65(ii), §10.65(iii), §10.68(iii).

Encodings:

BibTeX

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The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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38: 18.10 Integral Representations

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§18.10(i) Dirichlet–Mehler-Type Integral Representations

… ►Generalizations of (18.10.1) for $P_n^{(\alpha ,\beta )}$ are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)). … ►

18.10.6 $$L_n^{(\alpha )}\left(x^2\right)=\frac{2{(-1)}^n}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)n!}{\int }_0^{\mathrm{\infty }}{\int }_0^{\pi }{(x^2-r^2+2ixr\cos \varphi )}^n{\mathrm{e}}^{-r^2}{r}^{2\alpha +1}{(\sin \varphi )}^{2\alpha }d\varphi dr,$$ $\alpha >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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18.10.9 $$L_n^{(\alpha )}\left(x\right)=\frac{{\mathrm{e}}^x{x}^{-\frac{1}{2}\alpha }}{n!}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{n+\frac{1}{2}\alpha }{J}_{\alpha }\left(2\sqrt{xt}\right)dt,$$ $\alpha >-1$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $t$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.10.14

Referenced by:

§18.10(iv)

Permalink:

http://dlmf.nist.gov/18.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(iv), §18.10(iv), §18.10 and Ch.18

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39: 9.16 Physical Applications

… ►Extensive use is made of Airy functions in investigations in the theory of electromagnetic diffraction and radiowave propagation (Fock (1965)). … ►These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …

40: 16.8 Differential Equations

… ► … ►

§16.8(ii) The Generalized Hypergeometric Differential Equation

… ►We have the connection formula … ►In this reference it is also explained that in general when $q>1$ no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near $z=1$. … ►

§16.8(iii) Confluence of Singularities

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