expansions in series of Bessel functions

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31: 11.15 Approximations

§11.15(i) Expansions in Chebyshev Series

•

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{L}_{n}\left(x\right)$ , $0\leq\left|x\right|\leq 8$ , and $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , $x\geq 8$ , for $n=0,1$ ; $\int_{0}^{x}t^{-m}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}t^{-m}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ , $0\leq\left|x\right|\leq 8$ , $m=0,1$ and $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ , $x\geq 8$ ; the coefficients are to 20D.

â–º

•

MacLeod (1993) gives Chebyshev-series expansions for $\mathbf{L}_{0}\left(x\right)$ , $\mathbf{L}_{1}\left(x\right)$ , $0\leq x\leq 16$ , and $I_{0}\left(x\right)-\mathbf{L}_{0}\left(x\right)$ , $I_{1}\left(x\right)-\mathbf{L}_{1}\left(x\right)$ , $x\geq 16$ ; the coefficients are to 20D.

… â–º

•

Newman (1984) gives polynomial approximations for $\mathbf{H}_{n}\left(x\right)$ for $n=0,1$ , $0\leq x\leq 3$ , and rational-fraction approximations for $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ for $n=0,1$ , $x\geq 3$ . The maximum errors do not exceed 1.2×10â»â¸ for the former and 2.5×10â»â¸ for the latter.

32: 10.8 Power Series

§10.8 Power Series

â–ºFor

J_{\nu}\left(z\right)

see (10.2.2) and (10.4.1). When

\nu

is not an integer the corresponding expansions for

Y_{\nu}\left(z\right)

{H^{(1)}_{\nu}}\left(z\right)

, and

{H^{(2)}_{\nu}}\left(z\right)

are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8). … â–ºIn particular, … â–ºThe corresponding results for

{H^{(1)}_{n}}\left(z\right)

and

{H^{(2)}_{n}}\left(z\right)

are obtained via (10.4.3) with

\nu=n

. …

33: 3.6 Linear Difference Equations

… â–º

Example 1. Bessel Functions

… â–ºis satisfied by

J_{n}\left(1\right)

and

Y_{n}\left(1\right)

, where

J_{n}\left(x\right)

and

Y_{n}\left(x\right)

are the Bessel functions of the first kind. …Thus

Y_{n}\left(1\right)

is dominant and can be computed by forward recursion, whereas

J_{n}\left(1\right)

is recessive and has to be computed by backward recursion. … â–ºThus the asymptotic behavior of the particular solution

\mathbf{E}_{n}\left(1\right)

is intermediate to those of the complementary functions

J_{n}\left(1\right)

and

Y_{n}\left(1\right)

; moreover, the conditions for Olver’s algorithm are satisfied. We apply the algorithm to compute

\mathbf{E}_{n}\left(1\right)

to 8S for the range

n=1,2,\dots,10

, beginning with the value

\mathbf{E}_{0}\left(1\right)=-0.56865\;\,663

obtained from the Maclaurin series expansion (§11.10(iii)). …

34: 2.8 Differential Equations with a Parameter

… â–º

§2.8(iv) Case III: Simple Pole

… â–ºFor other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … â–ºFor a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. … â–ºFor further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. â–ºFor further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). …

35: Bibliography W

… â–º

E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

â“˜

External Links:: Document
Referenced by:: §2.11(vi), §2.11(vi).
Encodings:: BibTeX

… â–º

E. J. Weniger (2007) Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

â“˜

External Links:: ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt
Referenced by:: §2.10(i).
Encodings:: BibTeX

… â–º

A. D. Wheelon (1968) Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.

â“˜

External Links:: MathReview, ZentralBlatt
Referenced by:: §10.22(vi), §10.23(iv), §10.43(vi).
Encodings:: BibTeX

… â–º

R. Wong and J.-M. Zhang (1997) Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.

â“˜

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §18.34(iii).
Encodings:: BibTeX

… â–º

E. M. Wright (1935) The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

â“˜

External Links:: Document, ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

…

36: 14.15 Uniform Asymptotic Approximations

… â–ºIn other words, the convergent hypergeometric series expansions of

\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)

are also generalized (and uniform) asymptotic expansions as

\mu\to\infty

, with scale

\ifrac{1}{\Gamma\left(j+1+\mu\right)}

j=0,1,2,\dots

; compare §2.1(v). … â–ºHere

I

and

K

are the modified Bessel functions (§10.25(ii)). … â–ºFor the Bessel functions

J

and

Y

see §10.2(ii), and for the

\operatorname{env}

functions associated with

J

and

Y

see §2.8(iv). … â–ºFor convergent series expansions see Dunster (2004). … â–ºSee also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials

P_{n}\left(\cos\theta\right)

n\to\infty

with

\theta

fixed. …

37: Bibliography V

… â–º

H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

â“˜

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

… â–º

R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

â“˜

External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

… â–º

A. N. Vavreck and W. Thompson (1984) Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.

â“˜

External Links:: ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt
Referenced by:: §10.60(iii), §10.60(iii).
Encodings:: BibTeX

… â–º

H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

â“˜

External Links:: ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt
Referenced by:: §31.15(iii).
Encodings:: BibTeX

… â–º

H. Volkmer (2021) Fourier series representation of Ferrers function

{\sf P}

â“˜

External Links:: FullText
Referenced by:: (14.13.1).
Encodings:: BibTeX

…

38: 3.10 Continued Fractions

… â–ºEvery convergent, asymptotic, or formal series … â–ºFor several special functions the

S

-fractions are known explicitly, but in any case the coefficients

a_{n}

can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … â–ºFor applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … â–ºWe say that it is associated with the formal power series

f(z)

in (3.10.7) if the expansion of its

n

th convergent

C_{n}

in ascending powers of

z

, agrees with (3.10.7) up to and including the term in

z^{2n-1}

n=1,2,3,\dots

. … â–ºIn Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). …

39: Bibliography T

… â–º

I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

â“˜

External Links:: ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

… â–º

N. M. Temme (1979b) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

â“˜

External Links:: ISSN 0036-1410, Document, MathReview (E. RiekstiÅ†Å¡), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

… â–º

N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

â“˜

External Links:: ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt
Referenced by:: §13.8(iii), §13.8(iii).
Encodings:: BibTeX

… â–º

N. M. Temme (1994b) Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

â“˜

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.
Encodings:: BibTeX

… â–º

I. J. Thompson and A. R. Barnett (1986) Coulomb and Bessel functions of complex arguments and order. J. Comput. Phys. 64 (2), pp. 490–509.

â“˜

External Links:: ISSN 0021-9991, Document, MathReview (R. C. Varma), ZentralBlatt
Referenced by:: §10.74(v), §13.32(iii), §3.10(iii), §3.10(iii), §33.13, §33.23(vi), §33.26(iii), Ch.33, Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).
Encodings:: BibTeX

…

40: Bibliography C

… â–º

B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

â“˜

External Links:: ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §19.25(v), §22.15(ii).
Encodings:: BibTeX

… â–º

T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

â“˜

External Links:: MathReview (I. I. Hirschman, Jr.), ZentralBlatt
Referenced by:: §12.16.
Encodings:: BibTeX

… â–º

C. W. Clenshaw (1957) The numerical solution of linear differential equations in Chebyshev series. Proc. Cambridge Philos. Soc. 53 (1), pp. 134–149.

â“˜

External Links:: MathReview (L. Fox), ZentralBlatt
Referenced by:: §18.38(i), §3.11(ii).
Encodings:: BibTeX

… â–º

J. P. Coleman and A. J. Monaghan (1983) Chebyshev expansions for the Bessel function

J_{n}(z)

in the complex plane. Math. Comp. 40 (161), pp. 343–366.

â“˜

External Links:: ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

… â–º

R. M. Corless, D. J. Jeffrey, and D. E. Knuth (1997) A sequence of series for the Lambert

W

function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

â“˜

External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.
Encodings:: BibTeX

…