Hankel expansions

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31: Bibliography R

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High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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3rd item.

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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

External Links:

ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

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32: Bibliography

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Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:

MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§30.9(iii).

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Uniform asymptotic expansions for exponential integrals $E_n(x)$ and Bickley functions ${\mathrm{Ki}}_n(x)$ . ACM Trans. Math. Software 9 (4), pp. 467–479.

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ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

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Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 588; package TOMS)

Referenced by:

2nd item.

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Special value of the hypergeometric function ${}_3{}^{}F_2^{}$ and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

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ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§16.4(iii).

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33: Bibliography K

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Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.

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ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

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Addendum: “Hypergeometric expansions of Heun polynomials”. SIAM J. Math. Anal. 22 (6), pp. 1803.

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ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.5.

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Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

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Calculation of the complex zeros of Hankel functions and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 25 (11), pp. 1628–1643, 1741.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 25(6), pp. 26–36

External Links:

ISSN 0044-4669, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 4th item, 5th item.

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34: 10.49 Explicit Formulas

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10.49.6 $${𝗁}_n^{(1)}\left(z\right)={\mathrm{e}}^{\mathrm{i}z}\sum _{k=0}^n{\mathrm{i}}^{k-n-1}\frac{a_k(n+\frac{1}{2})}{z^{k+1}},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗁}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.1.16

Referenced by:

§10.52(ii)

Permalink:

http://dlmf.nist.gov/10.49.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(i), §10.49 and Ch.10

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10.49.7 $${𝗁}_n^{(2)}\left(z\right)={\mathrm{e}}^{-\mathrm{i}z}\sum _{k=0}^n{(-\mathrm{i})}^{k-n-1}\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗁}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.1.17 (corrected)

Permalink:

http://dlmf.nist.gov/10.49.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(i), §10.49 and Ch.10

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