in terms of Bessel functions of variable order

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21: 13.10 Integrals

… ►Other formulas of this kind can be constructed by inversion of the differentiation formulas given in §13.3(ii). … ►For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §§1.16 and 3.4.42–46, 4.4.45–47, 5.94–97). … ►For integral transforms in terms of Whittaker functions see §13.23(iv). … ►Generalized orthogonality integrals (33.14.13) and (33.14.15) can be expressed in terms of Kummer functions via the definitions in that section.

22: 10.21 Zeros

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§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►This subsection describes the distribution in $ℂ$ of the zeros of the principal branches of the Bessel functions of the second and third kinds, and their derivatives, in the case when the order is a positive integer $n$. … ►Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions. … ►

§10.21(xiv) $\nu$-Zeros

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23: 13.8 Asymptotic Approximations for Large Parameters

… ►For fixed $a$ and $z$ in $ℂ$ … ►uniformly with respect to bounded positive values of $x$ in each case. … ►where $C_{\nu }(a,\zeta )=\cos \left(\pi a\right)J_{\nu }\left(\zeta \right)+\sin \left(\pi a\right)Y_{\nu }\left(\zeta \right)$ and … ►where ${\mathrm{\Gamma }}^{\ast }\left(a\right)$ is the scaled gamma function defined in (5.11.3). These results follow from Temme (2022), which can also be used to obtain more terms in the expansions. …

24: Bibliography

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Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:

MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

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A subroutine package for Bessel functions of a complex argument and nonnegative order. Technical Report Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.

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External Links:

FullText

Referenced by:

1st item, B. Döring (1966).

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Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Software 12 (3), pp. 265–273.

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Notes:

Single and double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, Remark #1. Remark #2. GAMS (module 644; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item, 1st item, 2nd item, 1st item.

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Spherical Bessel functions $j_n$ and $y_n$ of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.

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Notes:

Single-precision Fortran, maximum accuracy 10S.

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ISSN 0010-4655, Document, Code

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1st item.

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25: Bibliography S

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A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§10.21(x).

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

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Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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Topics in Complex Function Theory. Vol. III: Abelian Functions and Modular Functions of Several Variables. Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience, [John Wiley & Sons, Inc], New York-London-Sydney.

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Notes:

Translated from the original German by E. Gottschling and M. Tretkoff. Reprinted by John Wiley & Sons, New York, 1989.

External Links:

ISBN 0-471-79090-7, MathReview, ZentralBlatt

Referenced by:

Ch.21.

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An inequality involving Bessel functions of argument nearly equal to their order. Proc. Amer. Math. Soc. 4 (6), pp. 858–859.

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ISSN 0002-9939, FullText, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§10.14.

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26: Bibliography L

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Inequalities and approximations for zeros of Bessel functions of small order. SIAM J. Math. Anal. 14 (2), pp. 383–388.

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ISSN 0036-1410, Document, MathReview (S. Ahmed), ZentralBlatt

Referenced by:

§10.21(iv).

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On Bessel functions of equal order and argument. Rend. Sem. Mat. Univ. Politec. Torino 50 (2), pp. 209–216 (1993).

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External Links:

ISSN 0373-1243, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.14.

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Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

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Monotonicity in terms of order of the zeros of the derivatives of Bessel functions. Proc. Amer. Math. Soc. 108 (2), pp. 387–389.

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External Links:

ISSN 0002-9939, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

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27: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►with the variable $\zeta$ defined implicitly by … ►For the functions $J_{2\mu }$, $Y_{2\mu }$, $H_{2\mu }^{(1)}$, and $H_{2\mu }^{(2)}$ see §10.2(ii), and for the $\mathrm{env}$ functions associated with $J_{2\mu }$ and $Y_{2\mu }$ see §2.8(iv). ►These approximations are proved in Dunster (1989). … ►These approximations are proved in Dunster (1989). … ►For a uniform asymptotic expansion in terms of Airy functions for $W_{\kappa ,\mu }\left(4\kappa x\right)$ when $\kappa$ is large and positive, $\mu$ is real with $|\mu |$ bounded, and $x\in [\delta ,\mathrm{\infty })$ see Olver (1997b, Chapter 11, Ex. 7.3). …

28: 11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

… ►If the series on the right-hand side of (11.6.1) is truncated after $m(\ge 0)$ terms, then the remainder term $R_m(z)$ is $O\left(z^{\nu -2m-1}\right)$. If $\nu$ is real, $z$ is positive, and $m+\frac{1}{2}-\nu \ge 0$, then $R_m(z)$ is of the same sign and numerically less than the first neglected term. … ►For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … ►

§11.6(ii) Large $|\nu |$, Fixed $z$

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29: Bibliography G

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

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Evaluation of the modified Bessel function of the third kind of imaginary orders. J. Comput. Phys. 175 (2), pp. 398–411.

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External Links:

ISSN 0021-9991, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.74(viii).

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Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. J. Comput. Appl. Math. 153 (1-2), pp. 225–234.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.45, §10.74(viii).

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Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.

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External Links:

ISSN 0098-3500, Document, GAMS (module 831; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item.

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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(viii).

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30: 18.16 Zeros

… ►Also, with $\rho$ defined as in (18.15.5) … ►Let $j_{\alpha ,m}$ be the $m$th positive zero of the Bessel function $J_{\alpha }\left(x\right)$ (§10.21(i)). Then … ►For three additional terms in this expansion see Gatteschi (2002). … ►Arrange them in decreasing order: …