leading%20coefficients

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21: 36.15 Methods of Computation

… ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. …

22: 9.16 Physical Applications

… ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)). …

23: Frank W. J. Olver

… ►He also served on the Editorial Boards of several of the leading journals devoted to mathematical and/or numerical analysis, including founding Managing Editor of the SIAM Journal on Mathematical Analysis. … ►According to DLMF Project Lead Daniel Lozier, “Olver’s encyclopedic knowledge of the field, his clear vision for mathematical exposition, his keen sense of the needs of practitioners, and his unfailing attention to detail were key to the success of that project. …

24: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$ , $5\leq s\leq 11$ , $11\leq s\leq 25$ , $25\leq s\leq 55$ . Precision is varied, with a maximum of 20S.

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$ , $k=2,3,4,5,8$ , for $0\leq s\leq 1$ (23D).

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Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta\left(s\right)$ for $0\leq s\leq 1$ (15D), $\zeta\left(s+1\right)$ for $0\leq s\leq 1$ (20D), and $\ln\xi\left(\tfrac{1}{2}+ix\right)$ (§25.4) for $-1\leq x\leq 1$ (20D). For errata see Piessens and Branders (1972).

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25: 28.35 Tables

§28.35 Tables

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Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

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Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$ , $\operatorname{Se}_{n+1}\left(x,q\right)$ for $n=0(1)5$ , $q=1(1)20$ , $x=0.1(.1)1$ ; 4D or 5D.

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National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$ , and $n=4(1)15$ with $q=0(2)100$ ; Fourier coefficients for $\operatorname{ce}_{n}\left(x,q\right)$ and $\operatorname{se}_{n}\left(x,q\right)$ for $n=0(1)15$ , $n=1(1)15$ , respectively, and various values of $q$ in the interval $[0,100]$ ; joining factors $g_{\mathit{e},n}(\sqrt{q})$ , $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$ . Precision is generally 8D.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n+1}\left(q\right)$ for $n=0(1)4$ , $q=0(1)50$ ; $n=0(1)20$ ( $a$ ’s) or 19 ( $b$ ’s), $q=1,3,5,10,15,25,50(50)200$ . Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , $n=0(1)7$ . Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $x=0(5^{\circ})90^{\circ}$ . Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $j=1,2$ , $x=0(.2)4$ . Precision is mostly 9S.

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26: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►The replacement of

f(t)

by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►In the sense of distributions, they can be written … ►Also, …Multiplication of these expansions leads to …

27: 8.11 Asymptotic Approximations and Expansions

… ►where … ►

8.11.9

b_{k}(\lambda)=\lambda(1-\lambda)b_{k-1}^{\prime}(\lambda)+(2k-1)\lambda b_{k-% 1}(\lambda).

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Symbols:: $k$ : nonnegative integer, $\lambda$ : parameter and $b_{k}(\lambda)$ : coefficients
Referenced by:: §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §8.11(iii), §8.11 and Ch.8

… ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. … ►

8.11.18

S_{n}(x)\sim\sum_{k=0}^{\infty}d_{k}(x)n^{-k},

n\to\infty

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $x$ : real variable, $k$ : nonnegative integer, $n$ : nonnegative integer, $S_{n}(x)$ : function and $d_{k}(x)$ : coefficients
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §8.11(v), §8.11 and Ch.8

►uniformly for

x\in(-\infty,1-\delta]

, with …

28: 10.75 Tables

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Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

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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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Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

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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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Olver (1960) tabulates $a^{\prime}_{n,m}$ , $\mathsf{j}_{n}\left(a^{\prime}_{n,m}\right)$ , $b^{\prime}_{n,m}$ , $\mathsf{y}_{n}\left(b^{\prime}_{n,m}\right)$ , $n=1(1)20$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ .

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29: 19.15 Advantages of Symmetry

… ►The function

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

(Carlson (1963)) reveals the full permutation symmetry that is partially hidden in

F_{D}

, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. …

30: 18.33 Polynomials Orthogonal on the Unit Circle

… ► … ►Askey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle. … ►with complex coefficients

c_{k}

and of a certain degree

n

define the reversed polynomial

p^{*}(z)

by …The Verblunsky coefficients (also called Schur parameters or reflection coefficients) are the coefficients

\alpha_{n}

in the Szegő recurrence relations … ►For

w(z)

as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure

\mu

in (18.33.17)) and with

\alpha_{n}

the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …