DLMF: Untitled Document

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Olver (1960) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(\tfrac{1}{2})20\tfrac{1}{2}$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ ; see §10.21(viii), and more fully Olver (1954).

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Abramowitz and Stegun (1964, Chapter 9) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $n=0(1)8$ , $m=1(1)20$ , 5D (10D for $n=0$ ), $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(1)8$ , $m=1(1)20$ , 5D (8D for $n=0$ ), $J_{0}\left(j_{0,m}x\right)$ , $m=1(1)5$ , $x=0(.02)1$ , 5D. Also included are the first 5 zeros of the functions $xJ_{1}\left(x\right)-\lambda J_{0}\left(x\right)$ , $J_{1}\left(x\right)-\lambda xJ_{0}\left(x\right)$ , $J_{0}\left(x\right)Y_{0}\left(\lambda x\right)-Y_{0}\left(x\right)J_{0}\left(% \lambda x\right)$ , $J_{1}\left(x\right)Y_{1}\left(\lambda x\right)-Y_{1}\left(x\right)J_{1}\left(% \lambda x\right)$ , $J_{1}\left(x\right)Y_{0}\left(\lambda x\right)-Y_{1}\left(x\right)J_{0}\left(% \lambda x\right)$ for various values of $\lambda$ and $\lambda^{-1}$ in the interval $[0,1]$ , 4–8D.

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Makinouchi (1966) tabulates all values of $j_{\nu,m}$ and $y_{\nu,m}$ in the interval $(0,100)$ , with at least 29S. These are for $\nu=0(1)5$ , 10, 20; $\nu=\tfrac{3}{2}$ , $\tfrac{5}{2}$ ; $\nu=m/n$ with $m=1(1)n-1$ and $n=3(1)8$ , except for $\nu=\tfrac{1}{2}$ .

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

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Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , 8D.

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Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

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B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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9.9.6

a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.7

\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)% \right),

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $V$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.8

a^{\prime}_{k}=-U\left(\tfrac{3}{8}\pi(4k-3)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ , $k$ : nonnegative integer and $U$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.9

\operatorname{Ai}\left(a^{\prime}_{k}\right)=(-1)^{k-1}W\left(\tfrac{3}{8}\pi(% 4k-3)\right).

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ , $k$ : nonnegative integer and $W$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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T. Pálmai and B. Apagyi (2011) Interlacing of positive real zeros of Bessel functions. J. Math. Anal. Appl. 375 (1), pp. 320–322.

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External Links:: ISSN 0022-247X, Document, FullText, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §10.21(i), §10.21(i).
Encodings:: BibTeX

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R. Parnes (1972) Complex zeros of the modified Bessel function

K_{n}(Z)

. Math. Comp. 26 (120), pp. 949–953.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

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R. Piessens (1984a) Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.

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External Links:: ISSN 0021-9991, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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R. Piessens (1990) On the computation of zeros and turning points of Bessel functions. Bull. Soc. Math. Grèce (N.S.) 31, pp. 117–122.

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External Links:: MathReview (Alan M. Cohen), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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§7.23(iv) Zeros

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Fettis et al. (1973) gives the first 100 zeros of $\operatorname{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$ , not for $\operatorname{erfc}z$ ), 11S.

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Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\operatorname{erf}z$ , 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$ , 8S.

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A. Laforgia and M. E. Muldoon (1983) Inequalities and approximations for zeros of Bessel functions of small order. SIAM J. Math. Anal. 14 (2), pp. 383–388.

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External Links:: ISSN 0036-1410, Document, MathReview (S. Ahmed), ZentralBlatt
Referenced by:: §10.21(iv).
Encodings:: BibTeX

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A. Laforgia and M. E. Muldoon (1988) Monotonicity properties of zeros of generalized Airy functions. Z. Angew. Math. Phys. 39 (2), pp. 267–271.

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External Links:: ISSN 0044-2275, Document, MathReview (Evangelos Ifantis), ZentralBlatt
Referenced by:: §9.13(i).
Encodings:: BibTeX

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P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.

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External Links:: ISSN 0021-9045, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §24.12(i), §24.9.
Encodings:: BibTeX

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§3.8(iv) Zeros of Polynomials

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§3.8(vi) Conditioning of Zeros

… ►The zeros of …Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. …

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L. Gatteschi (1987) New inequalities for the zeros of Jacobi polynomials. SIAM J. Math. Anal. 18 (6), pp. 1549–1562.

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External Links:: ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §18.16(ii), §18.16(vi).
Encodings:: BibTeX

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L. Gatteschi (1990) New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §13.9(i), §13.9(ii).
Encodings:: BibTeX

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W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

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A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

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S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

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M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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External Links:: ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt
Referenced by:: §31.15(ii).
Encodings:: BibTeX

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D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

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of%20zeros

1—10 of 33 matching pages

1: 10.75 Tables

2: 9.18 Tables

3: Bibliography D

4: 9.9 Zeros

5: Bibliography P

6: 7.23 Tables

§7.23(iv) Zeros

7: Bibliography L

8: 3.8 Nonlinear Equations

§3.8(iv) Zeros of Polynomials

§3.8(vi) Conditioning of Zeros

9: Bibliography G

10: Bibliography