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§10.75 Tables

Contents

§10.75(i) Introduction

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

  • British Association for the Advancement of Science (1937) tabulates J0(x), J1(x), x=0(.001)16(.01)25, 10D; Y0(x), Y1(x), x=0.01(.01)25, 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of Y0(x), Y1(x) for small values of x, as well as auxiliary functions to compute all four functions for large values of x.

  • Bickley et al. (1952) tabulates Jn(x), Yn(x) or xnYn(x), n=2(1)20, x=0(.01 or .1) 10(.1)25, 8D (for Jn(x)), 8S (for Yn(x) or xnYn(x)); Jn(x), Yn(x), n=0(1)20, x=0 or 0.1(.1)25, 10D (for Jn(x)), 10S (for Yn(x)).

  • Olver (1962) provides tables for the uniform asymptotic expansions given in §10.20(i), including ζ and (4ζ/(1-x2))14 as functions of x (=z) and the coefficients Ak(ζ), Bk(ζ), Ck(ζ), Dk(ζ) as functions of ζ. These enable Jν(νx), Yν(νx), Jν(νx), Yν(νx) to be computed to 10S when ν15, except in the neighborhoods of zeros.

  • The main tables in Abramowitz and Stegun (1964, Chapter 9) give J0(x) to 15D, J1(x), J2(x), Y0(x), Y1(x) to 10D, Y2(x) to 8D, x=0(.1)17.5; Yn(x)-(2/π)Jn(x)lnx, n=0,1, x=0(.1)2, 8D; Jn(x), Yn(x), n=3(1)9, x=0(.2)20, 5D or 5S; Jn(x), Yn(x), n=0(1)20(10)50,100, x=1,2,5,10,50,100, 10S; modulus and phase functions xMn(x), θn(x)-x, n=0,1,2, 1/x=0(.01)0.1, 8D.

  • Achenbach (1986) tabulates J0(x), J1(x), Y0(x), Y1(x), x=0(.1)8, 20D or 18–20S.

  • Zhang and Jin (1996, pp. 185–195) tabulates Jn(x), Jn(x), Yn(x), Yn(x), n=0(1)10(10)50,100, x=1, 5, 10, 25, 50, 100, 9S; Jn+α(x), Jn+α(x), Yn+α(x), Yn+α(x), n=0(1)5,10,30,50,100, α=14,13,12,23,34, x=1,5,10,50, 8S; real and imaginary parts of Jn+α(z), Jn+α(z), Yn+α(z), Yn+α(z), n=0(1)15,20(10)50,100, α=0,12, z=4+2i, 20+10i, 8S.

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

Real Zeros

  • British Association for the Advancement of Science (1937) tabulates j0,m, J1(j0,m), j1,m, J0(j1,m), m=1(1)150, 10D; y0,m, Y1(y0,m), y1,m, Y0(y1,m), m=1(1)50, 8D.

  • Olver (1960) tabulates jn,m, Jn(jn,m), jn,m, Jn(jn,m), yn,m, Yn(yn,m), yn,m, Yn(yn,m), n=0(12)2012, m=1(1)50, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n; see §10.21(viii), and more fully Olver (1954).

  • Morgenthaler and Reismann (1963) tabulates jn,m for n=21(1)51 and jn,m<100, 7-10S.

  • Abramowitz and Stegun (1964, Chapter 9) tabulates jn,m, Jn(jn,m), jn,m, Jn(jn,m), n=0(1)8, m=1(1)20, 5D (10D for n=0), yn,m, Yn(yn,m), yn,m, Yn(yn,m), n=0(1)8, m=1(1)20, 5D (8D for n=0), J0(j0,mx), m=1(1)5, x=0(.02)1, 5D. Also included are the first 5 zeros of the functions xJ1(x)-λJ0(x), J1(x)-λxJ0(x), J0(x)Y0(λx)-Y0(x)J0(λx), J1(x)Y1(λx)-Y1(x)J1(λx), J1(x)Y0(λx)-Y1(x)J0(λx) for various values of λ and λ-1 in the interval [0,1], 4–8D.

  • Abramowitz and Stegun (1964, Chapter 10) tabulates jν,m, Jν(jν,m), jν,m, Jν(jν,m), yν,m, Yν(yν,m), yν,m, Yν(yν,m), ν=12(1)1912, m=1(1)mν, where mν ranges from 8 at ν=12 down to 1 at ν=1912, 6–7D.

  • Makinouchi (1966) tabulates all values of jν,m and yν,m in the interval (0,100), with at least 29S. These are for ν=0(1)5, 10, 20; ν=32, 52; ν=m/n with m=1(1)n-1 and n=3(1)8, except for ν=12.

  • Döring (1971) tabulates the first 100 values of ν (>1) for which J-ν(x) has the double zero x=ν, 10D.

  • Heller (1976) tabulates j0,m, J1(j0,m), j1,m, J0(j1,m), j1,m, J1(j1,m) for m=1(1)100, 25D.

  • Wills et al. (1982) tabulates j0,m, j1,m, y0,m, y1,m for m=1(1)30, 35D.

  • Kerimov and Skorokhodov (1985c) tabulates 201 double zeros of J-ν′′(x), 10 double zeros of J-ν′′′(x), 101 double zeros of Y-ν(x), 201 double zeros of Y-ν′′(x), and 10 double zeros of Y-ν′′′(x), all to 8 or 9D.

  • Zhang and Jin (1996, pp. 196–198) tabulates jn,m, jn,m, yn,m, yn,m, n=0(1)3, m=1(1)10, 8D; the first five zeros of Jn(x)Yn(λx)-Jn(λx)Yn(x), Jn(x)Yn(λx)-Jn(λx)Yn(x), n=0,1,2, λ=1.1(.1)1.6,1.8,2(.5)5, 7D.

Complex Zeros

  • Abramowitz and Stegun (1964, p. 373) tabulates the three smallest zeros of Y0(z), Y1(z), Y1(z) in the sector 0<phzπ, together with the corresponding values of Y1(z), Y0(z), Y1(z), respectively, to 9D. (There is an error in the value of Y0(z) at the 3rd zero of Y1(z): the last four digits should be 2533; see Amos (1985).)

  • Döring (1966) tabulates all zeros of Y0(z), Y1(z), H0(1)(z), H1(1)(z), that lie in the sector |z|<158, |phz|π, to 10D. Some of the smaller zeros of Yn(z) and Hn(1)(z) for n=2,3,4,5,15 are also included.

  • Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of Yn(z) and Yn(z) for n=0(1)5, 8D.

  • Kerimov and Skorokhodov (1985b) tabulates 50 zeros of the principal branches of H0(1)(z) and H1(1)(z), 8D.

  • Kerimov and Skorokhodov (1987) tabulates 100 complex double zeros ν of Yν(ze-πi) and Hν(1)(ze-πi), 8D.

  • MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function J0(z)-iJ1(z), 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 Y0(z), Y1(z), Y1(z) and the corresponding values of Y1(z), Y0(z), Y1(z), respectively, 10D.

§10.75(iv) Integrals of Bessel Functions

  • Abramowitz and Stegun (1964, Chapter 11) tabulates 0xJ0(t)dt, 0xY0(t)dt, x=0(.1)10, 10D; 0xt-1(1-J0(t))dt, xt-1Y0(t)dt, x=0(.1)5, 8D.

  • Zhang and Jin (1996, p. 270) tabulates 0xJ0(t)dt, 0xt-1(1-J0(t))dt, 0xY0(t)dt, xt-1Y0(t)dt, x=0(.1)1(.5)20, 8D.

§10.75(v) Modified Bessel Functions and their Derivatives

  • British Association for the Advancement of Science (1937) tabulates I0(x), I1(x), x=0(.001)5, 7–8D; K0(x), K1(x), x=0.01(.01)5, 7–10D; e-xI0(x), e-xI1(x), exK0(x), exK1(x), x=5(.01)10(.1)20, 8D. Also included are auxiliary functions to facilitate interpolation of the tables of K0(x), K1(x) for small values of x.

  • Bickley et al. (1952) tabulates x-nIn(x) or e-xIn(x), xnKn(x) or exKn(x), n=2(1)20, x=0(.01 or .1) 10(.1) 20, 8S; In(x), Kn(x), n=0(1)20, x=0 or 0.1(.1)20, 10S.

  • Olver (1962) provides tables for the uniform asymptotic expansions given in §10.41(ii), including η and the coefficients Uk(p), Vk(p) as functions of p=(1+x2)-12. These enable Iν(νx), Kν(νx), Iν(νx), Kν(νx) to be computed to 10S when ν16.

  • The main tables in Abramowitz and Stegun (1964, Chapter 9) give e-xIn(x), exKn(x), n=0,1,2, x=0(.1)10(.2)20, 8D–10D or 10S; xe-xIn(x), (x/π) exKn(x), n=0,1,2, 1/x=0(.002)0.05; K0(x)+I0(x)lnx, x(K1(x)-I1(x)lnx), x=0(.1)2, 8D; e-xIn(x), exKn(x), n=3(1)9, x=0(.2)10(.5)20, 5S; In(x), Kn(x), n=0(1)20(10)50,100, x=1,2,5,10,50,100, 9–10S.

  • Achenbach (1986) tabulates I0(x), I1(x), K0(x), K1(x), x=0(.1)8, 19D or 19–21S.

  • Zhang and Jin (1996, pp. 240–250) tabulates In(x), In(x), Kn(x), Kn(x), n=0(1)10(10)50,100, x=1,5,10,25,50,100, 9S; In+α(x), In+α(x), Kn+α(x), Kn+α(x), n=0(1)5, 10, 30, 50, 100, α=14, 13, 12, 23, 34, x=1, 5, 10, 50, 8S; real and imaginary parts of In+α(z), In+α(z), Kn+α(z), Kn+α(z), n=0(1)15, 20(10)50, 100, α=0,12, z=4+2i,20+10i, 8S.

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

  • Parnes (1972) tabulates all zeros of the principal value of Kn(z), for n=2(1)10, 9D.

  • Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of Kn(z), for n=2(1)10, 29S.

  • Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of Kn(z) and Kn(z), for n=2(1)20, 9S.

  • Kerimov and Skorokhodov (1984c) tabulates all zeros of I-n-12(z) and I-n-12(z) in the sector 0phz12π for n=1(1)20, 9S.

  • Kerimov and Skorokhodov (1985b) tabulates all zeros of Kn(z) and Kn(z) in the sector -12π<phz32π for n=0(1)5, 8D.

§10.75(vii) Integrals of Modified Bessel Functions

  • Abramowitz and Stegun (1964, Chapter 11) tabulates e-x0xI0(t)dt, exxK0(t)dt, x=0(.1)10, 7D; e-x0xt-1(I0(t)-1)dt, xexxt-1K0(t)dt, x=0(.1)5, 6D.

  • Bickley and Nayler (1935) tabulates Kin(x)10.43(iii)) for n=1(1)16, x=0(.05)0.2(.1) 2, 3, 9D.

  • Zhang and Jin (1996, p. 271) tabulates e-x0xI0(t)dt, e-x0xt-1(I0(t)-1)dt, exxK0(t)dt, xexxt-1K0(t)dt, x=0(.1)1(.5)20, 8D.

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

For the notation see §10.45.

  • Žurina and Karmazina (1967) tabulates K~ν(x) for ν=0.01(.01)10, x=0.1(.1)10.2, 7S.

  • Rappoport (1979) tabulates the real and imaginary parts of K12+iτ(x) for τ=0.01(.01)10, x=0.1(.2)9.5, 7S.

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

  • The main tables in Abramowitz and Stegun (1964, Chapter 10) give jn(x), yn(x) n=0(1)8, x=0(.1)10, 5–8S; jn(x), yn(x) n=0(1)20(10)50, 100, x=1,2,5,10,50,100, 10S; in(1)(x), kn(x), n=0,1,2, x=0(.1)5, 4–9D; in(1)(x), kn(x), n=0(1)20(10)50, 100, x=1,2,5,10,50,100, 10S. (For the notation see §10.1 and §10.47(ii).)

  • Zhang and Jin (1996, pp. 296–305) tabulates jn(x), jn(x), yn(x), yn(x), in(1)(x), in(1)(x), kn(x), kn(x), n=0(1)10(10)30, 50, 100, x=1, 5, 10, 25, 50, 100, 8S; xjn(x), (xjn(x)), xyn(x), (xyn(x)) (Riccati–Bessel functions and their derivatives), n=0(1)10(10)30, 50, 100, x=1, 5, 10, 25, 50, 100, 8S; real and imaginary parts of jn(z), jn(z), yn(z), yn(z), in(1)(z), in(1)(z), kn(z), kn(z), n=0(1)15, 20(10)50, 100, z=4+2i, 20+10i, 8S. (For the notation replace j,y,i,k by j, y, i(1), k, respectively.)

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

For the notation see §10.58.

  • Olver (1960) tabulates an,m, jn(an,m), bn,m, yn(bn,m), n=1(1)20, m=1(1)50, 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n.

§10.75(xi) Kelvin Functions and their Derivatives

  • Young and Kirk (1964) tabulates bernx, beinx, kernx, keinx, n=0,1, x=0(.1)10, 15D; bernx, beinx, kernx, keinx, modulus and phase functions Mn(x), θn(x), Nn(x), ϕn(x), n=0,1,2, x=0(.01)2.5, 8S, and n=0(1)10, x=0(.1)10, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for n=0(1)10 for small values of x. (Concerning the phase functions see §10.68(iv).)

  • Abramowitz and Stegun (1964, Chapter 9) tabulates bernx, beinx, kernx, keinx, n=0,1, x=0(.1)5, 9–10D; xn(kernx+(bernx)(lnx)), xn(keinx+(beinx)(lnx)), n=0,1, x=0(.1)1, 9D; modulus and phase functions Mn(x), θn(x), Nn(x), ϕn(x), n=0,1, x=0(.2)7, 6D; xe-x/2Mn(x), θn(x)-(x/2), xex/2Nn(x), ϕn(x)+(x/2), n=0,1, 1/x=0(.01)0.15, 5D.

  • Zhang and Jin (1996, p. 322) tabulates berx, berx, beix, beix, kerx, kerx, keix, keix, x=0(1)20, 7S.

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

  • Zhang and Jin (1996, p. 323) tabulates the first 20 real zeros of berx, berx, beix, beix, kerx, kerx, keix, keix, 8D.