Bessel-function%20expansion

SearchAdvancedHelp

(0.004 seconds)

1—10 of 19 matching pages

1: Bibliography D

… ►

Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

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

Referenced by:

§10.41(iii), §2.8(i).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§10.49(ii), §18.34(iii).

Encodings:

BibTeX

…

2: 10.75 Tables

… ►

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_0\left(x\right)$ to 15D, $J_1\left(x\right)$, $J_2\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$ to 10D, $Y_2\left(x\right)$ to 8D, $x=0(.1)17.5$; $Y_n\left(x\right)-(2/\pi )J_n\left(x\right)\ln x$, $n=0,1$, $x=0(.1)2$, 8D; $J_n\left(x\right)$, $Y_n\left(x\right)$, $n=3(1)9$, $x=0(.2)20$, 5D or 5S; $J_n\left(x\right)$, $Y_n\left(x\right)$, $n=0(1)20(10)50,100$, $x=1,2,5,10,50,100$, 10S; modulus and phase functions $\sqrt{x}{M}_n\left(x\right)$, ${\theta }_n\left(x\right)-x$, $n=0,1,2$, $1/x=0(.01)0.1$, 8D.

… ►

Olver (1960) tabulates $j_{n,m}$, $J_n^{\prime }\left(j_{n,m}\right)$, $j_{n,m}^{\prime }$, $J_n\left(j_{n,m}^{\prime }\right)$, $y_{n,m}$, $Y_n^{\prime }\left(y_{n,m}\right)$, $y_{n,m}^{\prime }$, $Y_n\left(y_{n,m}^{\prime }\right)$, $n=0(\frac{1}{2})20⁤\frac{1}{2}$, $m=1(1)50$, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to \mathrm{\infty }$; see §10.21(viii), and more fully Olver (1954).

… ►

Abramowitz and Stegun (1964, Chapter 9) tabulates $j_{n,m}$, $J_n^{\prime }\left(j_{n,m}\right)$, $j_{n,m}^{\prime }$, $J_n\left(j_{n,m}^{\prime }\right)$, $n=0(1)8$, $m=1(1)20$, 5D (10D for $n=0$), $y_{n,m}$, $Y_n^{\prime }\left(y_{n,m}\right)$, $y_{n,m}^{\prime }$, $Y_n\left(y_{n,m}^{\prime }\right)$, $n=0(1)8$, $m=1(1)20$, 5D (8D for $n=0$), $J_0\left(j_{0,m}x\right)$, $m=1(1)5$, $x=0(.02)1$, 5D. Also included are the first 5 zeros of the functions $x{J}_1\left(x\right)-\lambda J_0\left(x\right)$, $J_1\left(x\right)-\lambda x{J}_0\left(x\right)$, $J_0\left(x\right)Y_0\left(\lambda x\right)-Y_0\left(x\right)J_0\left(\lambda x\right)$, $J_1\left(x\right)Y_1\left(\lambda x\right)-Y_1\left(x\right)J_1\left(\lambda x\right)$, $J_1\left(x\right)Y_0\left(\lambda x\right)-Y_1\left(x\right)J_0\left(\lambda x\right)$ for various values of $\lambda$ and ${\lambda }^{-1}$ in the interval $[0,1]$, 4–8D.

… ►

§10.75(iv) Integrals of Bessel Functions

… ►

Zhang and Jin (1996, pp. 296–305) tabulates ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗒}_n^{\prime }\left(x\right)$, ${𝗂}_n^{(1)}\left(x\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(x\right)$, ${𝗄}_n\left(x\right)$, ${𝗄}_n^{\prime }\left(x\right)$, $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; $x{𝗃}_n\left(x\right)$, ${(x{𝗃}_n\left(x\right))}^{\prime }$, $x{𝗒}_n\left(x\right)$, ${(x{𝗒}_n\left(x\right))}^{\prime }$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; real and imaginary parts of ${𝗃}_n\left(z\right)$, ${𝗃}_n^{\prime }\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗒}_n^{\prime }\left(z\right)$, ${𝗂}_n^{(1)}\left(z\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(z\right)$, ${𝗄}_n\left(z\right)$, ${𝗄}_n^{\prime }\left(z\right)$, $n=0(1)15$, 20(10)50, 100, $z=4+2\mathrm{i}$, $20+10\mathrm{i}$, 8S. (For the notation replace $j,y,\mathrm{i},k$ by $𝗃$, $𝗒$, ${𝗂}^{(1)}$, $𝗄$, respectively.)

…

3: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►

§30.9(ii) Oblate Spheroidal Wave Functions

… ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►

§30.9(iii) Other Approximations and Expansions

…

4: Bibliography B

… ►

Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

ⓘ

External Links:

ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.45.

Encodings:

BibTeX

… ►

A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

… ►

A short table of the functions ${\mathrm{Ki}}_n(x)$, from $n=1$ to $n=16$ . Phil. Mag. Series 7 20, pp. 343–347.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§10.43(iii), 2nd item.

Encodings:

BibTeX

… ►

Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

…

5: Bibliography S

… ►

Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

… ►

Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

ⓘ

External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

►

Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

ⓘ

External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

ⓘ

Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

ⓘ

Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

Encodings:

BibTeX

…

6: Bibliography R

… ►

A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

… ►

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

ⓘ

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

Applied Bessel Functions. Dover Publications Inc., New York.

ⓘ

Notes:

Reprint of original Blackie & Son Limited, London, 1946.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.73(iii).

Encodings:

BibTeX

►

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.

ⓘ

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).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.23(ii).

Encodings:

BibTeX

…

7: 6.20 Approximations

… ►

Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

… ►

MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

… ►

Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\mathrm{Ein}\left(ax\right)$, $\mathrm{Si}\left(ax\right)$, and $\mathrm{Cin}\left(ax\right)$ for $-1\le x\le 1$, $a\in ℂ$. The coefficients are given in terms of series of Bessel functions.

… ►

Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If the scheme can be used in backward direction.

►

§6.20(iii) Padé-Type and Rational Expansions

…

8: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

►

§11.6(i) Large $|z|$, Fixed $\nu$

… ►

§11.6(ii) Large $|\nu |$, Fixed $z$

… ► … ►Here …

9: Bibliography P

… ►

Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

►

A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

Encodings:

BibTeX

… ►

Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

ⓘ

External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

… ►

Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

… ►

A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

ⓘ

External Links:

ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

10: Bibliography C

… ►

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

ⓘ

External Links:

MathReview (I. I. Hirschman, Jr.), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

… ►

Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

ⓘ

External Links:

ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

… ►

The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

ⓘ

External Links:

MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

… ►

Chebyshev expansions for the Bessel function $J_n(z)$ in the complex plane. Math. Comp. 40 (161), pp. 343–366.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

BibTeX

… ►

Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

…