relation to zeros

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21: 19.14 Reduction of General Elliptic Integrals

… ►In (19.14.1)–(19.14.3) both the integrand and

\cos\phi

are assumed to be nonnegative. … ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. …

22: Bibliography O

… ►

A. M. Odlyzko (1987) On the distribution of spacings between zeros of the zeta function. Math. Comp. 48 (177), pp. 273–308.

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External Links:: ISSN 0025-5718, FullText, Author’s Website, MathReview (S. L. Segal), ZentralBlatt
Referenced by:: §25.18(ii).
Encodings:: BibTeX

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O. M. Ogreid and P. Osland (1998) Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

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External Links:: ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt
Referenced by:: §25.8.
Encodings:: BibTeX

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F. W. J. Olver (1950) A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.

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External Links:: Document, MathReview (J. Todd), ZentralBlatt
Referenced by:: §10.21(ii).
Encodings:: BibTeX

►

F. W. J. Olver (1951) A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.

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External Links:: Document, MathReview (J. Todd), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

… ►

F. W. J. Olver (Ed.) (1960) Bessel Functions. Part III: Zeros and Associated Values. Royal Society Mathematical Tables, Volume 7, Cambridge University Press, Cambridge-New York.

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External Links:: ISBN 0-521-06153-9, MathReview (L. Fox), ZentralBlatt
Referenced by:: §10.21(vi), §10.21(viii), §10.58, §10.74(vi), 1st item, 2nd item.
Encodings:: BibTeX

…

23: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

… ►

§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

… ► … ►

§19.25(vii) Hypergeometric Function

… ►

24: 33.23 Methods of Computation

… ►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii

\rho

and

r

, respectively, and may be used to compute the regular and irregular solutions. … ►

§33.23(iv) Recurrence Relations

►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer

\ell

, provided that the recurrence is carried out in a stable direction (§3.6). … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …

25: 1.11 Zeros of Polynomials

§1.11 Zeros of Polynomials

►

§1.11(i) Division Algorithm

… ►Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. … ►A similar relation holds for the changes in sign of the coefficients of

f(-z)

, and hence for the number of negative zeros of

f(z)

. … ►Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. …

26: Errata

… ►

Additions

Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F\left(a,a;a+1;\tfrac{1}{2}\right)$ , (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

… ►

Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

… ►

Equation (9.7.2)

Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_{k}=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}u_{k-1}$ was added to Equation (9.7.2).

… ►

Chapter 25 Zeta and Related Functions

A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

… ►

Subsection 1.2(i)

The condition for (1.2.2), (1.2.4), and (1.2.5) was corrected. These equations are true only if $n$ is a positive integer. Previously $n$ was allowed to be zero.

Reported 2011-08-10 by Michael Somos.

…

27: 25.16 Mathematical Applications

… ►which is related to the Riemann zeta function by …where the sum is taken over the nontrivial zeros

\rho

\zeta\left(s\right)

. ►The prime number theorem (27.2.3) is equivalent to the statement … ►

H\left(s\right)

is analytic for

\Re s>1

, and can be extended meromorphically into the half-plane

\Re s>-2k

for every positive integer

k

by use of the relations … ►Related results are: …

28: 35.6 Confluent Hypergeometric Functions of Matrix Argument

… ►

35.6.4

{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\operatorname{% etr}\left(\mathbf{T}\mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+% 1)}\left|\mathbf{I}-\mathbf{X}\right|^{b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},

\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(ii), §35.6 and Ch.35

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35.6.6

\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left|\mathbf{T}\right|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left|\mathbf{X}\right|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left|\mathbf{T}-\mathbf{X}\right|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}},

\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)

… ►

§35.6(iii) Relations to Bessel Functions of Matrix Argument

►

35.6.9

\lim_{a\to\infty}{{}_{1}F_{1}}\left({a\atop\nu+\frac{1}{2}(m+1)};-a^{-1}% \mathbf{T}\right)=\frac{A_{\nu}\left(\mathbf{T}\right)}{A_{\nu}\left(% \boldsymbol{{0}}\right)}.

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Symbols:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.6.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(iii), §35.6 and Ch.35

►

35.6.10

\lim_{a\to\infty}\Gamma_{m}\left(a\right)\Psi\left(a+\nu;\nu+\tfrac{1}{2}(m+1)% ;a^{-1}\mathbf{T}\right)=B_{\nu}\left(\mathbf{T}\right).

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Symbols:: $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (second kind), $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(iii), §35.6 and Ch.35

…

29: Bibliography

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V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:: ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt
Referenced by:: (25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.
Encodings:: BibTeX

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W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:: FullText, MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §24.14(ii).
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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T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

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F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

…

30: Bibliography M

… ►

S. Makinouchi (1966) 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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Referenced by:: 6th item.
Encodings:: BibTeX

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I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and

k

-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:: ISSN 0022-2488, Document, Link, MathReview Entry
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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M. Micu (1968) Recursion relations for the

3

j

symbols. Nuclear Physics A 113 (1), pp. 215–220.

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External Links:: Document
Referenced by:: §34.3(iii).
Encodings:: BibTeX

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S. C. Milne (1985c) A new symmetry related to

\mathit{SU}(n)

for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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M. E. Muldoon (1981) 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).
Encodings:: BibTeX

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