in terms of Bessel functions of fixed order

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11: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

… ►The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$, $C_{1,0}=0$, and … ►The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). ►

§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

12: 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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13: 11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

… ►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 $z$

… ►For fixed $\lambda (>1)$ …and for fixed $\lambda$ $(>0)$ …

14: Bibliography K

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Calculation of modified Bessel functions in a complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(3), pp. 15–24

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(iv).

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On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

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Notes:

In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(i), §10.74(vi).

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An improvement of the remainder term in the divisor problem. Mat. Zametki 6, pp. 545–554 (Russian).

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Notes:

In Russian. English translation: Math. Notes 6(1969), no. 5, pp. 784–791.

External Links:

Document, MathReview (A. I. Vinogradov), ZentralBlatt

Referenced by:

§27.11, Erratum (V1.1.8) for Section 27.11.

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Hypergeometric functions of $2\times 2$ matrix argument are expressible in terms of Appel’s functions $F_4$ . Proc. Amer. Math. Soc. 70 (1), pp. 39–42.

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External Links:

FullText, MathReview (P. N. Rathie), ZentralBlatt

Referenced by:

§35.7(ii).

Encodings:

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ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.

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Notes:

Double-precision Fortran, maximum accuracy 25S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

2nd item.

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15: 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).

Encodings:

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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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External Links:

ISSN 1095-7170, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(v).

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

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External Links:

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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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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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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External Links:

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

Referenced by:

§10.14.

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16: 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).

Encodings:

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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).

Encodings:

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17: Errata

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18.34.1 $$y_n(x;a)={}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-n,n+a-1}{-};-\frac{x}{2})={\left(n+a-1\right)}_n{\left(\frac{x}{2}\right)}^n{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{-2n-a+2};\frac{2}{x})=n!{\left(-\frac{1}{2}x\right)}^n{L}_n^{(1-a-2n)}\left(2x^{-1}\right)={\left(\frac{1}{2}x\right)}^{1-\frac{1}{2}a}{\mathrm{e}}^{1/x}{W}_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right)$$

This equation was updated to include the definition of Bessel polynomials in terms of Laguerre polynomials and the Whittaker confluent hypergeometric function.

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18.34.2

$y_n(x)=y_n(x;2)=2{\pi }^{-1}{x}^{-1}{\mathrm{e}}^{1/x}{𝗄}_n\left(x^{-1}\right),$

${\theta }_n(x)=x^n{y}_n(x^{-1})=2{\pi }^{-1}{x}^{n+1}{\mathrm{e}}^x{𝗄}_n\left(x\right)$

This equation was updated to include definitions in terms of the modified spherical Bessel function of the second kind.

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Originally it was stated incorrectly that $\rho$ was fixed. This has been corrected to state that $\eta \rho$ is fixed.

Reported by Ian Thompson on 2018-05-17

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The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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Originally was expressed in term of asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

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