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10 Bessel Functions Computation 10.74 Methods of Computation 10.76 Approximations

§10.75 Tables

Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.75

Contents

§10.75(i) Introduction
§10.75(ii) Bessel Functions and their Derivatives
§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives
§10.75(iv) Integrals of Bessel Functions
§10.75(v) Modified Bessel Functions and their Derivatives
§10.75(vi) Zeros of Modified Bessel Functions and their Derivatives
§10.75(vii) Integrals of Modified Bessel Functions
§10.75(viii) Modified Bessel Functions of Imaginary or Complex Order
§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives
§10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions
§10.75(xi) Kelvin Functions and their Derivatives
§10.75(xii) Zeros of Kelvin Functions and their Derivatives

§10.75(i) Introduction

Permalink:: http://dlmf.nist.gov/10.75.i

Comprehensive listings and descriptions of tables of the functions treated in this chapter are provided in Bateman and Archibald (1944), Lebedev and Fedorova (1960), Fletcher et al. (1962), and Luke (1975, §9.13.2). Only a few of the more comprehensive of these early tables are included in the listings in the following subsections. Also, for additional listings of tables pertaining to complex arguments see Babushkina et al. (1997).

§10.75(ii) Bessel Functions and their Derivatives

Permalink:: http://dlmf.nist.gov/10.75.ii

British Association for the Advancement of Science (1937) tabulates $\mathop{J_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{J_{{1}}\/}\nolimits\!\left(x\right)$ , , 10D; $\mathop{Y_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(x\right)$ , , 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $\mathop{Y_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(x\right)$ for small values of , as well as auxiliary functions to compute all four functions for large values of .
Bickley et al. (1952) tabulates $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ or $x^{n}\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ , , (.01 or .1) , 8D (for $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ ), 8S (for $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ or $x^{n}\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ ); $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ , , or , 10D (for $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ ), 10S (for $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ ).
Olver (1962) provides tables for the uniform asymptotic expansions given in §10.20(i), including $\zeta$ and $(\ifrac{4\zeta}{(1-x^{2})})^{{\frac{1}{4}}}$ as functions of and the coefficients $A_{k}(\zeta)$ , $B_{k}(\zeta)$ , $C_{k}(\zeta)$ , $D_{k}(\zeta)$ as functions of $\zeta$ . These enable $\mathop{J_{{\nu}}\/}\nolimits\!\left(\nu x\right)$ , $\mathop{Y_{{\nu}}\/}\nolimits\!\left(\nu x\right)$ , ${\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)$ , ${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)$ to be computed to 10S when $\nu\geq 15$ , except in the neighborhoods of zeros.
The main tables in Abramowitz and Stegun (1964, Chapter 9) give $\mathop{J_{{0}}\/}\nolimits\!\left(x\right)$ to 15D, $\mathop{J_{{1}}\/}\nolimits\!\left(x\right)$ , $\mathop{J_{{2}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(x\right)$ to 10D, $\mathop{Y_{{2}}\/}\nolimits\!\left(x\right)$ to 8D, ; $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)-(2/\pi)\mathop{J_{{n}}\/}\nolimits% \!\left(x\right)\mathop{\ln\/}\nolimits x$ , , , 8D; $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ , , , 5D or 5S; $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ , , , 10S; modulus and phase functions $\sqrt{x}\mathop{M_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\theta_{{n}}\/}\nolimits\!\left(x\right)-x$ , , $\ifrac{1}{x}=0(.01)0.1$ , 8D.
Achenbach (1986) tabulates $\mathop{J_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{J_{{1}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(x\right)$ , , 20D or 18–20S.
Zhang and Jin (1996, pp. 185–195) tabulates $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{J_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{Y_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , , , 5, 10, 25, 50, 100, 9S; $\mathop{J_{{n+\alpha}}\/}\nolimits\!\left(x\right)$ , ${\mathop{J_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{Y_{{n+\alpha}}\/}\nolimits\!\left(x\right)$ , ${\mathop{Y_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , , $\alpha=\tfrac{1}{4},\tfrac{1}{3},\tfrac{1}{2},\tfrac{2}{3},\tfrac{3}{4}$ , , 8S; real and imaginary parts of $\mathop{J_{{n+\alpha}}\/}\nolimits\!\left(z\right)$ , ${\mathop{J_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{Y_{{n+\alpha}}\/}\nolimits\!\left(z\right)$ , ${\mathop{Y_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , , $\alpha=0,\tfrac{1}{2}$ , , , 8S.

§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

Keywords:: zeros of Bessel functions (including derivatives)
Referenced by:: §10.21(ix)
Permalink:: http://dlmf.nist.gov/10.75.iii

¶ Real Zeros

British Association for the Advancement of Science (1937) tabulates $\mathop{j_{{0,m}}\/}\nolimits$ , $\mathop{J_{{1}}\/}\nolimits\!\left(\mathop{j_{{0,m}}\/}\nolimits\right)$ , $\mathop{j_{{1,m}}\/}\nolimits$ , $\mathop{J_{{0}}\/}\nolimits\!\left(\mathop{j_{{1,m}}\/}\nolimits\right)$ , , 10D; $\mathop{y_{{0,m}}\/}\nolimits$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(\mathop{y_{{0,m}}\/}\nolimits\right)$ , $\mathop{y_{{1,m}}\/}\nolimits$ , $\mathop{Y_{{0}}\/}\nolimits\!\left(\mathop{y_{{1,m}}\/}\nolimits\right)$ , , 8D.
Olver (1960) tabulates $\mathop{j_{{n,m}}\/}\nolimits$ , ${\mathop{J_{{n}}\/}\nolimits^{{\prime}}}\!\left(\mathop{j_{{n,m}}\/}\nolimits\right)$ , $\mathop{{j^{{\prime}}_{{n,m}}}\/}\nolimits$ , $\mathop{J_{{n}}\/}\nolimits\!\left(\mathop{{j^{{\prime}}_{{n,m}}}\/}\nolimits\right)$ , $\mathop{y_{{n,m}}\/}\nolimits$ , ${\mathop{Y_{{n}}\/}\nolimits^{{\prime}}}\!\left(\mathop{y_{{n,m}}\/}\nolimits\right)$ , $\mathop{{y^{{\prime}}_{{n,m}}}\/}\nolimits$ , $\mathop{Y_{{n}}\/}\nolimits\!\left(\mathop{{y^{{\prime}}_{{n,m}}}\/}\nolimits\right)$ , $n=0(\tfrac{1}{2})20\tfrac{1}{2}$ , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ ; see §10.21(viii), and more fully Olver (1954).
Morgenthaler and Reismann (1963) tabulates $\mathop{{j^{{\prime}}_{{n,m}}}\/}\nolimits$ for and $\mathop{{j^{{\prime}}_{{n,m}}}\/}\nolimits{<100}$ , 7-10S.
Abramowitz and Stegun (1964, Chapter 9) tabulates $\mathop{j_{{n,m}}\/}\nolimits$ , ${\mathop{J_{{n}}\/}\nolimits^{{\prime}}}\!\left(\mathop{j_{{n,m}}\/}\nolimits\right)$ , $\mathop{{j^{{\prime}}_{{n,m}}}\/}\nolimits$ , $\mathop{J_{{n}}\/}\nolimits\!\left(\mathop{{j^{{\prime}}_{{n,m}}}\/}\nolimits\right)$ , , , 5D (10D for ), $\mathop{y_{{n,m}}\/}\nolimits$ , ${\mathop{Y_{{n}}\/}\nolimits^{{\prime}}}\!\left(\mathop{y_{{n,m}}\/}\nolimits\right)$ , $\mathop{{y^{{\prime}}_{{n,m}}}\/}\nolimits$ , $\mathop{Y_{{n}}\/}\nolimits\!\left(\mathop{{y^{{\prime}}_{{n,m}}}\/}\nolimits\right)$ , , , 5D (8D for ), $\mathop{J_{{0}}\/}\nolimits\!\left(\mathop{j_{{0,m}}\/}\nolimits x\right)$ , , , 5D. Also included are the first 5 zeros of the functions $x\mathop{J_{{1}}\/}\nolimits\!\left(x\right)-\lambda\mathop{J_{{0}}\/}% \nolimits\!\left(x\right)$ , $\mathop{J_{{1}}\/}\nolimits\!\left(x\right)-\lambda x\mathop{J_{{0}}\/}% \nolimits\!\left(x\right)$ , $\mathop{J_{{0}}\/}\nolimits\!\left(x\right)\mathop{Y_{{0}}\/}\nolimits\!\left(% \lambda x\right)-\mathop{Y_{{0}}\/}\nolimits\!\left(x\right)\mathop{J_{{0}}\/}% \nolimits\!\left(\lambda x\right)$ , $\mathop{J_{{1}}\/}\nolimits\!\left(x\right)\mathop{Y_{{1}}\/}\nolimits\!\left(% \lambda x\right)-\mathop{Y_{{1}}\/}\nolimits\!\left(x\right)\mathop{J_{{1}}\/}% \nolimits\!\left(\lambda x\right)$ , $\mathop{J_{{1}}\/}\nolimits\!\left(x\right)\mathop{Y_{{0}}\/}\nolimits\!\left(% \lambda x\right)-\mathop{Y_{{1}}\/}\nolimits\!\left(x\right)\mathop{J_{{0}}\/}% \nolimits\!\left(\lambda x\right)$ for various values of $\lambda$ and $\lambda^{{-1}}$ in the interval , 4–8D.
Abramowitz and Stegun (1964, Chapter 10) tabulates $\mathop{j_{{\nu,m}}\/}\nolimits$ , ${\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\mathop{j_{{\nu,m}}\/}% \nolimits\right)$ , $\mathop{{j^{{\prime}}_{{\nu,m}}}\/}\nolimits$ , $\mathop{J_{{\nu}}\/}\nolimits\!\left(\mathop{{j^{{\prime}}_{{\nu,m}}}\/}% \nolimits\right)$ , $\mathop{y_{{\nu,m}}\/}\nolimits$ , ${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\mathop{y_{{\nu,m}}\/}% \nolimits\right)$ , $\mathop{{y^{{\prime}}_{{\nu,m}}}\/}\nolimits$ , $\mathop{Y_{{\nu}}\/}\nolimits\!\left(\mathop{{y^{{\prime}}_{{\nu,m}}}\/}% \nolimits\right)$ , $\nu=\tfrac{1}{2}(1)19\tfrac{1}{2}$ , $m=1(1)m_{\nu}$ , where $m_{\nu}$ ranges from 8 at $\nu=\tfrac{1}{2}$ down to 1 at $\nu=19\tfrac{1}{2}$ , 6–7D.
Makinouchi (1966) tabulates all values of $\mathop{j_{{\nu,m}}\/}\nolimits$ and $\mathop{y_{{\nu,m}}\/}\nolimits$ in the interval , with at least 29S. These are for $\nu=0(1)5$ , 10, 20; $\nu=\tfrac{3}{2}$ , $\tfrac{5}{2}$ ; $\nu=m/n$ with and , except for $\nu=\tfrac{1}{2}$ .
Döring (1971) tabulates the first 100 values of $\nu$ for which ${\mathop{J_{{-\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ has the double zero $x=\nu$ , 10D.
Heller (1976) tabulates $\mathop{j_{{0,m}}\/}\nolimits$ , $\mathop{J_{{1}}\/}\nolimits\!\left(\mathop{j_{{0,m}}\/}\nolimits\right)$ , $\mathop{j_{{1,m}}\/}\nolimits$ , $\mathop{J_{{0}}\/}\nolimits\!\left(\mathop{j_{{1,m}}\/}\nolimits\right)$ , $\mathop{{j^{{\prime}}_{{1,m}}}\/}\nolimits$ , $\mathop{J_{{1}}\/}\nolimits\!\left(\mathop{{j^{{\prime}}_{{1,m}}}\/}\nolimits\right)$ for , 25D.
Wills et al. (1982) tabulates $\mathop{j_{{0,m}}\/}\nolimits$ , $\mathop{j_{{1,m}}\/}\nolimits$ , $\mathop{y_{{0,m}}\/}\nolimits$ , $\mathop{y_{{1,m}}\/}\nolimits$ for , 35D.
Kerimov and Skorokhodov (1985c) tabulates 201 double zeros of ${\mathop{J_{{-\nu}}\/}\nolimits^{{\prime\prime}}}\!\left(x\right)$ , 10 double zeros of ${\mathop{J_{{-\nu}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(x\right)$ , 101 double zeros of ${\mathop{Y_{{-\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , 201 double zeros of ${\mathop{Y_{{-\nu}}\/}\nolimits^{{\prime\prime}}}\!\left(x\right)$ , and 10 double zeros of ${\mathop{Y_{{-\nu}}\/}\nolimits^{{\prime\prime\prime}}}\!\left(x\right)$ , all to 8 or 9D.
Zhang and Jin (1996, pp. 196–198) tabulates $\mathop{j_{{n,m}}\/}\nolimits$ , $\mathop{{j^{{\prime}}_{{n,m}}}\/}\nolimits$ , $\mathop{y_{{n,m}}\/}\nolimits$ , $\mathop{{y^{{\prime}}_{{n,m}}}\/}\nolimits$ , , , 8D; the first five zeros of $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)\mathop{Y_{{n}}\/}\nolimits\!\left(% \lambda x\right)-\mathop{J_{{n}}\/}\nolimits\!\left(\lambda x\right)\mathop{Y_% {{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{J_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right){\mathop{Y_{{n}}\/}% \nolimits^{{\prime}}}\!\left(\lambda x\right)-{\mathop{J_{{n}}\/}\nolimits^{{% \prime}}}\!\left(\lambda x\right){\mathop{Y_{{n}}\/}\nolimits^{{\prime}}}\!% \left(x\right)$ , , $\lambda=1.1(.1)1.6,1.8,2(.5)5$ , 7D.

¶ Complex Zeros

Keywords:: Bessel functions, Hankel functions

Abramowitz and Stegun (1964, p. 373) tabulates the three smallest zeros of $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ , ${\mathop{Y_{{1}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ in the sector $0<\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$ , together with the corresponding values of $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ , respectively, to 9D. (There is an error in the value of $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)$ at the 3rd zero of $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ : the last four digits should be 2533; see Amos (1985).)
Döring (1966) tabulates all zeros of $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ , $\mathop{{H^{{(1)}}_{{0}}}\/}\nolimits\!\left(z\right)$ , $\mathop{{H^{{(1)}}_{{1}}}\/}\nolimits\!\left(z\right)$ , that lie in the sector , $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$ , to 10D. Some of the smaller zeros of $\mathop{Y_{{n}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ for are also included.
Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of $\mathop{Y_{{n}}\/}\nolimits\!\left(z\right)$ and ${\mathop{Y_{{n}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ for , 8D.
Kerimov and Skorokhodov (1985b) tabulates 50 zeros of the principal branches of $\mathop{{H^{{(1)}}_{{0}}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{{(1)}}_{{1}}}\/}\nolimits\!\left(z\right)$ , 8D.
Kerimov and Skorokhodov (1987) tabulates 100 complex double zeros $\nu$ of ${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(ze^{{-\pi i}}\right)$ and ${\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(ze^{{-\pi i}}\right)$ , 8D.
MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $\mathop{J_{{0}}\/}\nolimits\!\left(z\right)-i\mathop{J_{{1}}\/}\nolimits\!% \left(z\right)$ , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).
Zhang and Jin (1996, p. 199) tabulates the real and imaginary parts of the first 15 conjugate pairs of complex zeros of $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ , ${\mathop{Y_{{1}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ and the corresponding values of $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{1}}\/}\nolimits\!\left(z\right)$ , respectively, 10D.

§10.75(iv) Integrals of Bessel Functions

Keywords:: Bessel functions, integrals
Permalink:: http://dlmf.nist.gov/10.75.iv

Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}\mathop{J_{{0}}\/}\nolimits\!\left(t\right)dt$ , $\int_{0}^{x}\mathop{Y_{{0}}\/}\nolimits\!\left(t\right)dt$ , , 10D; $\int_{0}^{x}t^{{-1}}(1-\mathop{J_{{0}}\/}\nolimits\!\left(t\right))dt$ , $\int_{x}^{\infty}t^{{-1}}\mathop{Y_{{0}}\/}\nolimits\!\left(t\right)dt$ , , 8D.
Zhang and Jin (1996, p. 270) tabulates $\int_{0}^{x}\mathop{J_{{0}}\/}\nolimits\!\left(t\right)dt$ , $\int_{0}^{x}t^{{-1}}(1-\mathop{J_{{0}}\/}\nolimits\!\left(t\right))dt$ , $\int_{0}^{x}\mathop{Y_{{0}}\/}\nolimits\!\left(t\right)dt$ , $\int_{x}^{\infty}t^{{-1}}\mathop{Y_{{0}}\/}\nolimits\!\left(t\right)dt$ , , 8D.

§10.75(v) Modified Bessel Functions and their Derivatives

Keywords:: integrals of modified Bessel functions, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.75.v

British Association for the Advancement of Science (1937) tabulates $\mathop{I_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{I_{{1}}\/}\nolimits\!\left(x\right)$ , , 7–8D; $\mathop{K_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{K_{{1}}\/}\nolimits\!\left(x\right)$ , , 7–10D; $e^{{-x}}\mathop{I_{{0}}\/}\nolimits\!\left(x\right)$ , $e^{{-x}}\mathop{I_{{1}}\/}\nolimits\!\left(x\right)$ , $e^{x}\mathop{K_{{0}}\/}\nolimits\!\left(x\right)$ , $e^{x}\mathop{K_{{1}}\/}\nolimits\!\left(x\right)$ , , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $\mathop{K_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{K_{{1}}\/}\nolimits\!\left(x\right)$ for small values of .
Bickley et al. (1952) tabulates $x^{{-n}}\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ or $e^{{-x}}\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ , $x^{n}\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ or $e^{x}\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ , , (.01 or .1) 10(.1) 20, 8S; $\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ , , or , 10S.
Olver (1962) provides tables for the uniform asymptotic expansions given in §10.41(ii), including $\eta$ and the coefficients $U_{k}(p)$ , $V_{k}(p)$ as functions of $p=(1+x^{2})^{{-\frac{1}{2}}}$ . These enable $\mathop{I_{{\nu}}\/}\nolimits\!\left(\nu x\right)$ , $\mathop{K_{{\nu}}\/}\nolimits\!\left(\nu x\right)$ , ${\mathop{I_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)$ , ${\mathop{K_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu x\right)$ to be computed to 10S when $\nu\geq 16$ .
The main tables in Abramowitz and Stegun (1964, Chapter 9) give $e^{{-x}}\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ , $e^{x}\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ , , , 8D–10D or 10S; $\sqrt{x}e^{{-x}}\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ , $(\sqrt{x}/\pi)$ $e^{x}\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ , , ; $\mathop{K_{{0}}\/}\nolimits\!\left(x\right)+\mathop{I_{{0}}\/}\nolimits\!\left% (x\right)\mathop{\ln\/}\nolimits x$ , $x(\mathop{K_{{1}}\/}\nolimits\!\left(x\right)-\mathop{I_{{1}}\/}\nolimits\!% \left(x\right)\mathop{\ln\/}\nolimits x)$ , , 8D; $e^{{-x}}\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ , $e^{x}\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ , , , 5S; $\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ , , , 9–10S.
Achenbach (1986) tabulates $\mathop{I_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{I_{{1}}\/}\nolimits\!\left(x\right)$ , $\mathop{K_{{0}}\/}\nolimits\!\left(x\right)$ , $\mathop{K_{{1}}\/}\nolimits\!\left(x\right)$ , , 19D or 19–21S.
Zhang and Jin (1996, pp. 240–250) tabulates $\mathop{I_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{I_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{K_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{K_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , , , 9S; $\mathop{I_{{n+\alpha}}\/}\nolimits\!\left(x\right)$ , ${\mathop{I_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{K_{{n+\alpha}}\/}\nolimits\!\left(x\right)$ , ${\mathop{K_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , , 10, 30, 50, 100, $\alpha=\tfrac{1}{4}$ , $\tfrac{1}{3}$ , $\tfrac{1}{2}$ , $\tfrac{2}{3}$ , $\tfrac{3}{4}$ , , 5, 10, 50, 8S; real and imaginary parts of $\mathop{I_{{n+\alpha}}\/}\nolimits\!\left(z\right)$ , ${\mathop{I_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{K_{{n+\alpha}}\/}\nolimits\!\left(z\right)$ , ${\mathop{K_{{n+\alpha}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , , 20(10)50, 100, $\alpha=0,\tfrac{1}{2}$ , , 8S.

§10.75(vi) Zeros of Modified Bessel Functions and their Derivatives

Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.75.vi

Parnes (1972) tabulates all zeros of the principal value of $\mathop{K_{{n}}\/}\nolimits\!\left(z\right)$ , for , 9D.
Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of $\mathop{K_{{n}}\/}\nolimits\!\left(z\right)$ , for , 29S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $\mathop{K_{{n}}\/}\nolimits\!\left(z\right)$ and ${\mathop{K_{{n}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , for , 9S.
Kerimov and Skorokhodov (1984c) tabulates all zeros of $\mathop{I_{{-n-\frac{1}{2}}}\/}\nolimits\!\left(z\right)$ and ${\mathop{I_{{-n-\frac{1}{2}}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ in the sector $0\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{2}\pi$ for , 9S.
Kerimov and Skorokhodov (1985b) tabulates all zeros of $\mathop{K_{{n}}\/}\nolimits\!\left(z\right)$ and ${\mathop{K_{{n}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ in the sector $-\tfrac{1}{2}\pi<\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2}\pi$ for , 8D.

§10.75(vii) Integrals of Modified Bessel Functions

Keywords:: Bessel functions, integrals
Permalink:: http://dlmf.nist.gov/10.75.vii

Abramowitz and Stegun (1964, Chapter 11) tabulates $e^{{-x}}\int_{0}^{x}\mathop{I_{{0}}\/}\nolimits\!\left(t\right)dt$ , $e^{x}\int_{x}^{\infty}\mathop{K_{{0}}\/}\nolimits\!\left(t\right)dt$ , , 7D; $e^{{-x}}\int_{0}^{x}t^{{-1}}(\mathop{I_{{0}}\/}\nolimits\!\left(t\right)-1)dt$ , $xe^{x}\int_{x}^{\infty}t^{{-1}}\mathop{K_{{0}}\/}\nolimits\!\left(t\right)dt$ , , 6D.
Bickley and Nayler (1935) tabulates $\mathop{\mathrm{Ki}_{{n}}\/}\nolimits\!\left(x\right)$ (§10.43(iii)) for , 2, 3, 9D.
Zhang and Jin (1996, p. 271) tabulates $e^{{-x}}\int_{0}^{x}\mathop{I_{{0}}\/}\nolimits\!\left(t\right)dt$ , $e^{{-x}}\int_{0}^{x}t^{{-1}}(\mathop{I_{{0}}\/}\nolimits\!\left(t\right)-1)dt$ , $e^{x}\int_{x}^{\infty}\mathop{K_{{0}}\/}\nolimits\!\left(t\right)dt$ , $xe^{x}\int_{x}^{\infty}t^{{-1}}\mathop{K_{{0}}\/}\nolimits\!\left(t\right)dt$ , , 8D.

§10.75(viii) Modified Bessel Functions of Imaginary or Complex Order

Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.75.viii

For the notation see §10.45.

Žurina and Karmazina (1967) tabulates $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ for $\nu=0.01(.01)10$ , , 7S.
Rappoport (1979) tabulates the real and imaginary parts of $\mathop{K_{{\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right)$ for $\tau=0.01(.01)10$ , , 7S.

§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives

Keywords:: derivatives, spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.75.ix

The main tables in Abramowitz and Stegun (1964, Chapter 10) give $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right)$ , , 5–8S; $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right)$ , 100, , 10S; $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(x\right)$ , , , 4–9D; $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(x\right)$ , , 100, , 10S. (For the notation see §10.1 and §10.47(ii).)
Zhang and Jin (1996, pp. 296–305) tabulates $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathsf{y}_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(x\right)$ , ${\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(x\right)$ , ${\mathop{\mathsf{k}_{{n}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ , , 50, 100, , 5, 10, 25, 50, 100, 8S; $x\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right)$ , $(x\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(x\right))^{{\prime}}$ , $x\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right)$ , $(x\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(x\right))^{{\prime}}$ (Riccati–Bessel functions and their derivatives), , 50, 100, , 5, 10, 25, 50, 100, 8S; real and imaginary parts of $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathsf{y}_{{n}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , ${\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ , ${\mathop{\mathsf{k}_{{n}}\/}\nolimits^{{\prime}}}\!\left(z\right)$ , , 20(10)50, 100, , , 8S. (For the notation replace by $\mathop{\mathsf{j}\/}\nolimits$ , $\mathop{\mathsf{y}\/}\nolimits$ , $\mathop{{\mathsf{i}^{{(1)}}}\/}\nolimits$ , $\mathop{\mathsf{k}\/}\nolimits$ , respectively.)

§10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions

Permalink:: http://dlmf.nist.gov/10.75.x

For the notation see §10.58.

Olver (1960) tabulates $a^{{\prime}}_{{n,m}}$ , $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(a^{{\prime}}_{{n,m}}\right)$ , $b^{{\prime}}_{{n,m}}$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(b^{{\prime}}_{{n,m}}\right)$ , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ .

§10.75(xi) Kelvin Functions and their Derivatives

Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.75.xi

Young and Kirk (1964) tabulates $\mathop{\mathrm{ber}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{bei}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{ker}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{kei}_{{n}}\/}\nolimits x$ , , , 15D; $\mathop{\mathrm{ber}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{bei}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{ker}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{kei}_{{n}}\/}\nolimits x$ , modulus and phase functions $\mathop{M_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\theta_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{N_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\phi_{{n}}\/}\nolimits\!\left(x\right)$ , , , 8S, and , , 7S. Also included are auxiliary functions to facilitate interpolation of the tables for for small values of . (Concerning the phase functions see §10.68(iv).)
Abramowitz and Stegun (1964, Chapter 9) tabulates $\mathop{\mathrm{ber}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{bei}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{ker}_{{n}}\/}\nolimits x$ , $\mathop{\mathrm{kei}_{{n}}\/}\nolimits x$ , , , 9–10D; $x^{n}(\mathop{\mathrm{ker}_{{n}}\/}\nolimits x+(\mathop{\mathrm{ber}_{{n}}\/}% \nolimits x)(\mathop{\ln\/}\nolimits x))$ , $x^{n}(\mathop{\mathrm{kei}_{{n}}\/}\nolimits x+(\mathop{\mathrm{bei}_{{n}}\/}% \nolimits x)(\mathop{\ln\/}\nolimits x))$ , , , 9D; modulus and phase functions $\mathop{M_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\theta_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{N_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\phi_{{n}}\/}\nolimits\!\left(x\right)$ , , , 6D; $\sqrt{x}e^{{-x/\sqrt{2}}}\mathop{M_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\theta_{{n}}\/}\nolimits\!\left(x\right)-(x/\sqrt{2})$ , $\sqrt{x}e^{{x/\sqrt{2}}}\mathop{N_{{n}}\/}\nolimits\!\left(x\right)$ , $\mathop{\phi_{{n}}\/}\nolimits\!\left(x\right)+(x/\sqrt{2})$ , , , 5D.
Zhang and Jin (1996, p. 322) tabulates $\mathop{\mathrm{ber}\/}\nolimits x$ , ${\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x$ , $\mathop{\mathrm{bei}\/}\nolimits x$ , ${\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}}x$ , $\mathop{\mathrm{ker}\/}\nolimits x$ , ${\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x$ , $\mathop{\mathrm{kei}\/}\nolimits x$ , ${\mathop{\mathrm{kei}\/}\nolimits^{{\prime}}}x$ , , 7S.

§10.75(xii) Zeros of Kelvin Functions and their Derivatives

Permalink:: http://dlmf.nist.gov/10.75.xii

Zhang and Jin (1996, p. 323) tabulates the first 20 real zeros of $\mathop{\mathrm{ber}\/}\nolimits x$ , ${\mathop{\mathrm{ber}\/}\nolimits^{{\prime}}}x$ , $\mathop{\mathrm{bei}\/}\nolimits x$ , ${\mathop{\mathrm{bei}\/}\nolimits^{{\prime}}}x$ , $\mathop{\mathrm{ker}\/}\nolimits x$ , ${\mathop{\mathrm{ker}\/}\nolimits^{{\prime}}}x$ , $\mathop{\mathrm{kei}\/}\nolimits x$ , ${\mathop{\mathrm{kei}\/}\nolimits^{{\prime}}}x$ , 8D.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.74 Methods of Computation 10.76 Approximations