relation to modified Bessel functions

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11: 18.15 Asymptotic Approximations

… ►where $J_{\nu }\left(z\right)$ is the Bessel function (§10.2(ii)), and … ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►

In Terms of Bessel Functions

… ►Here $J_{\nu }\left(z\right)$ denotes the Bessel function (§10.2(ii)), $\mathrm{env}{J}_{\nu }\left(z\right)$ denotes its envelope (§2.8(iv)), and $\delta$ is again an arbitrary small positive constant. … ►With $\mu =\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$, which is related to the Hermite polynomials via …

12: 10.51 Recurrence Relations and Derivatives

§10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, or ${𝗁}_n^{(2)}\left(z\right)$. … ►

§10.51(ii) Modified Functions

►Let $g_n(z)$ denote ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, or ${(-1)}^n$ ${𝗄}_n\left(z\right)$. Then …

13: Bibliography G

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Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. 420 (1), pp. 373–386.

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ISSN 0022-247X, Document, Link, MathReview (Nihat Yagmur), ZentralBlatt

Referenced by:

§10.37, §10.37.

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On the computation of modified Bessel function ratios. Math. Comp. 32 (143), pp. 865–875.

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ISSN 0025-5718, FullText, MathReview (James M. Horner), ZentralBlatt

Referenced by:

§10.74(v).

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Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.

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ISSN 0006-3835, Document, MathReview (Khélifa Trimèche), ZentralBlatt

Referenced by:

§10.74(iii), §9.17(iii).

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Some integrals involving three modified Bessel functions. I. J. Math. Phys. 27 (3), pp. 682–687.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

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Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

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ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.6(iii).

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14: 11.4 Basic Properties

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§11.4(ii) Inequalities

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§11.4(iv) Expansions in Series of Bessel Functions

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§11.4(v) Recurrence Relations and Derivatives

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§11.4(vi) Derivatives with Respect to Order

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§11.4(vii) Zeros

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15: 10.9 Integral Representations

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Poisson’s and Related Integrals

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Schläfli’s and Related Integrals

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Mehler–Sonine and Related Integrals

… ►For the function $I_{\nu }$ see §10.25(ii). … ►For the function $K_0$ see §10.25(ii). …

16: Bibliography B

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

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

Referenced by:

§10.45.

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

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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.

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MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

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Algorithm 484: Evaluation of the modified Bessel functions K0(Z) and K1(Z) for complex arguments. Comm. ACM 17 (9), pp. 524–526.

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Notes:

Double-precision Fortran code, maximum accuracy 13S.

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ISSN 0001-0782, Document, Code

Referenced by:

1st item.

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17: 13.8 Asymptotic Approximations for Large Parameters

… ►For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … ►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). … ►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 … ►For generalizations in which $z$ is also allowed to be large see Temme and Veling (2022).

18: 10.29 Recurrence Relations and Derivatives

§10.29 Recurrence Relations and Derivatives

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§10.29(i) Recurrence Relations

►With ${𝒵}_{\nu }\left(z\right)$ defined as in §10.25(ii), … ►For results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). ►

§10.29(ii) Derivatives

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19: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation … ►

§11.10(vi) Relations to Other Functions

… ► … ►

§11.10(ix) Recurrence Relations and Derivatives

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20: Bibliography S

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A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.

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

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7th item.

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

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Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374 (2), pp. 516–528.

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ISSN 0022-247X, Document, FullText, MathReview (Beyza Caliskan Aslan), ZentralBlatt

Referenced by:

§10.37, §10.37.

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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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On the calculation of complex zeros of the modified Bessel function of the second kind. Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.

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Notes:

English translation in Soviet Math. Dokl. 31(1), pp. 78–81

External Links:

ISSN 0002-3264, MathReview (I. F. Kushnirchuk), ZentralBlatt

Referenced by:

§10.74(vi).

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