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21—30 of 219 matching pages

21: William P. Reinhardt

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… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

23: Peter L. Walker

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20 Theta Functions

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24: 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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25: 20.10 Integrals

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26: Bibliography K

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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S. Kida (1981) A vortex filament moving without change of form. J. Fluid Mech. 112, pp. 397–409.

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External Links:: ISSN 0022-1120, Document, MathReview, ZentralBlatt
Referenced by:: §19.35(ii).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

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27: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

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k_{1}=\frac{k}{\sqrt{1+k^{2}}},

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k_{1}k_{1}^{\prime}=\frac{k}{1+k^{2}},

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22.17.7

\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k_{1}^{\prime},k_{% 1}\right),

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Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathrm{i}$ : imaginary unit, $z$ : complex, $k$ : modulus, $k_{1}$ : change of variable and $k_{1}^{\prime}$ : change of variable
A&S Ref:: 16.10.3 (modified)
Permalink:: http://dlmf.nist.gov/22.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►For proofs of these results and further information see Walker (2003).

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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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.

29: 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. …

30: 12.10 Uniform Asymptotic Expansions for Large Parameter

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12.10.33

\mathsf{A}_{s+1}(\tau)=-4\tau^{2}(\tau+1)^{2}\frac{\mathrm{d}}{\mathrm{d}\tau}% \mathsf{A}_{s}(\tau)-\frac{1}{4}\int_{0}^{\tau}\left(20u^{2}+20u+3\right)% \mathsf{A}_{s}(u)\,\mathrm{d}u,

s=0,1,2,\dots

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Defines:: $\mathsf{A}_{s}(\tau)$ : coefficients (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $s$ : nonnegative integer and $\tau$
Permalink:: http://dlmf.nist.gov/12.10.E33
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(vi), §12.10 and Ch.12

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\mathsf{A}_{1}(\tau)=-\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9),

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12.10.40

\phi(\zeta)=\left(\frac{\zeta}{t^{2}-1}\right)^{\frac{1}{4}}.

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Defines:: $\phi(\zeta)$ : function (locally)
Symbols:: $\zeta$ : change of variable
Referenced by:: §12.10(vii)
Permalink:: http://dlmf.nist.gov/12.10.E40
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.41

t=1+w-\tfrac{1}{10}w^{2}+\tfrac{11}{350}w^{3}-\tfrac{823}{63000}w^{4}+\tfrac{1% \;50653}{242\;55000}w^{5}+\cdots,

|\zeta|<\left(\tfrac{3}{4}\pi\right)^{\frac{2}{3}}.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter and $\zeta$ : change of variable
Referenced by:: §12.11(iii), §12.11(iii)
Permalink:: http://dlmf.nist.gov/12.10.E41
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.45

\chi(\zeta)=\frac{\phi^{\prime}(\zeta)}{\phi(\zeta)}=\frac{1-2t(\phi(\zeta))^{% 6}}{4\zeta}.

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Defines:: $\chi(\zeta)$ (locally)
Symbols:: $\zeta$ : change of variable and $\phi(\zeta)$ : function
Permalink:: http://dlmf.nist.gov/12.10.E45
Encodings:: pMML, png, TeX
See also:: Annotations for §12.10(vii), §12.10 and Ch.12

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