relation to Bessel functions

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31: 10.68 Modulus and Phase Functions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. … ►Thus this reference gives ${\varphi }_1\left(0\right)=\frac{5}{4}\pi$ (Eq. …10)), and ${lim}_{x\to \mathrm{\infty }}({\varphi }_1\left(x\right)+(x/\sqrt{2}))=-\frac{5}{8}\pi$ (Eqs. …However, numerical tabulations show that if the second of these equations applies and ${\varphi }_1\left(x\right)$ is continuous, then ${\varphi }_1\left(0\right)=-\frac{3}{4}\pi$; compare Abramowitz and Stegun (1964, p. 433).

32: 10.51 Recurrence Relations and Derivatives

§10.51 Recurrence Relations and Derivatives

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§10.51(i) Unmodified Functions

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

33: 10 Bessel Functions

Chapter 10 Bessel Functions

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34: Bibliography H

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

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Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§35.1, §35.2, §35.3(ii), §35.5(i), §35.5(ii), §35.5(iii), §35.6(i), §35.6(ii), §35.6(iii), §35.6(iv), §35.7(i), §35.7(ii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), Ch.35, Notations P.

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On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.

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ISSN 1364-5021, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.20(ii).

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Bessel function expansions of Coulomb wave functions. J. Math. Phys. 26 (4), pp. 656–659.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§33.10(ii), §33.10(iii), §33.14(iv), §33.9(ii), §33.9(ii).

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35: 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 …

36: 10.14 Inequalities; Monotonicity

§10.14 Inequalities; Monotonicity

… ►For monotonicity properties of $J_{\nu }\left(\nu \right)$ and $J_{\nu }^{\prime }\left(\nu \right)$ see Lorch (1992). …For a related bound for $Y_{\nu }\left(\nu x\right)$ see Siegel and Sleator (1954). … ►

Kapteyn’s Inequality

… ►For inequalities for the function $\mathrm{\Gamma }\left(\nu +1\right){(2/x)}^{\nu }{J}_{\nu }\left(x\right)$ with $\nu >-\frac{1}{2}$ see Neuman (2004). …

37: Bibliography M

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Zeros of Bessel functions $J_{\nu }(x)$ and $Y_{\nu }(x)$ accurate to twenty-nine significant digits. Technology Reports of the Osaka University 16 (685), pp. 1–44.

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6th item.

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On the roots of the Bessel and certain related functions. Ann. of Math. 9 (1-6), pp. 23–30.

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ISSN 0003-486X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.21(x).

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.

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External Links:

MathReview (S. Ahmed), ZentralBlatt

Referenced by:

§10.21(iv).

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38: 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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39: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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Stieltjes Fractions

… ►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►This forward algorithm achieves efficiency and stability in the computation of the convergents $C_n=A_n/B_n$, and is related to the forward series recurrence algorithm. …

40: 10.58 Zeros

§10.58 Zeros

►For $n\ge 0$ the $m$th positive zeros of ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, and ${𝗒}_n^{\prime }\left(x\right)$ are denoted by $a_{n,m}$, $a_{n,m}^{\prime }$, $b_{n,m}$, and $b_{n,m}^{\prime }$, respectively, except that for $n=0$ we count $x=0$ as the first zero of ${𝗃}_0^{\prime }\left(x\right)$. … ►

$a_{n,m}=j_{n+\frac{1}{2},m},$

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$b_{n,m}=y_{n+\frac{1}{2},m},$

… ►However, there are no simple relations that connect the zeros of the derivatives. …