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31: 33.23 Methods of Computation

… ►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … ►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). …On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►

§33.23(vi) Other Numerical Methods

►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …

32: 3.12 Mathematical Constants

… ►Other constants that appear in the DLMF include the base

e

of natural logarithms … ►For historical and other information see Finch (2003).

33: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iv) Relations to Other Functions

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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34: 7.22 Methods of Computation

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§7.22(v) Other References

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35: 10.36 Other Differential Equations

§10.36 Other Differential Equations

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36: 12.21 Software

… ►References to research software that is available in other ways is listed separately. … ►For other software see the Software Index.

37: 18.41 Tables

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§18.41(iii) Other Tables

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38: 36.14 Other Physical Applications

§36.14 Other Physical Applications

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39: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

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18.34.1

y_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,n+a-1\atop-};-\frac{x}{2}\right)=% {\left(n+a-1\right)_{n}}\left(\frac{x}{2}\right)^{n}{{}_{1}F_{1}}\left({-n% \atop-2n-a+2};\frac{2}{x}\right)=n!\left(-\tfrac{1}{2}x\right)^{n}L^{(1-a-2n)}% _{n}\left(2x^{-1}\right)=\left(\tfrac{1}{2}x\right)^{1-\frac{1}{2}a}{\mathrm{e% }}^{1/x}W_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right).

ⓘ

Defines:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Proved:: Grosswald (1978, p.38, Theorem 1); Combine (18.5.12) and items (i), (v), (vii) of the mentioned theorem for $b=2$ (proved)
Referenced by:: (18.34.2), (18.34.7_1), §18.34(i), §18.34(i), Erratum (V1.2.0) for Equation (18.34.1)
Permalink:: http://dlmf.nist.gov/18.34.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.
See also:: Annotations for §18.34(i), §18.34 and Ch.18

… ►Other notations in use are given by …where

\mathsf{k}_{n}

is a modified spherical Bessel function (10.49.9), and … ►

§18.34(iii) Other Properties

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40: Need Help?

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