Abel%20summability

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22: Staff

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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

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William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

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23: 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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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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Zhang and Jin (1996, p. 322) tabulates $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , $x=0(1)20$ , 7S.

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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.

24: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

25: Errata

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Section 1.13

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$ -argument Wronskian is given by $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$ , where $1\leq j,k\leq n$ . Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$ -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$ th-order differential equations. A reference to Ince (1926, §5.2) was added.

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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26: 1.2 Elementary Algebra

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§1.2(iv) Means

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27: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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28: Foreword

… ►November 20, 2009 …

29: 13.30 Tables

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Zhang and Jin (1996, pp. 411–423) tabulates $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ for $a=-5(.5)5$ , $b=0.5(.5)5$ , and $x=0.1,1,5,10,20,30$ , 8S (for $M\left(a,b,x\right)$ ) and 7S (for $U\left(a,b,x\right)$ ).

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30: 28.16 Asymptotic Expansions for Large $q$

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28.16.1

\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $h$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.16 and Ch.28

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