DLMF: Untitled Document

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Abramowitz and Stegun (1964) tabulates: $\zeta\left(n\right)$ , $n=2,3,4,\dots$ , 20D (p. 811); $\operatorname{Li}_{2}\left(1-x\right)$ , $x=0(.01)0.5$ , 9D (p. 1005); $f(\theta)$ , $\theta=15^{\circ}(1^{\circ})30^{\circ}(2^{\circ})90^{\circ}(5^{\circ})180^{\circ}$ , $f(\theta)+\theta\ln\theta$ , $\theta=0(1^{\circ})15^{\circ}$ , 6D (p. 1006). Here $f(\theta)$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ , $x=-5(.05)25$ , to 12S.

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… ►Spenceley and Spenceley (1947) tabulates

\operatorname{sn}\left(Kx,k\right)

,

\operatorname{cn}\left(Kx,k\right)

,

\operatorname{dn}\left(Kx,k\right)

,

\operatorname{am}\left(Kx,k\right)

,

\mathcal{E}\left(Kx,k\right)

for

\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}

and

x=0\left(\tfrac{1}{90}\right)1

to 12D, or 12 decimals of a radian in the case of

\operatorname{am}\left(Kx,k\right)

. … ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

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M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.12(i).
Encodings:: BibTeX

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H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:: For corrigendum see same journal v. 90 (1998), no. 2, p. 265
External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §5.6(i).
Encodings:: BibTeX

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G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1990) Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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G. E. Andrews and A. Berkovich (1998) A trinomial analogue of Bailey’s lemma and

N=2

superconformal invariance. Comm. Math. Phys. 192 (2), pp. 245–260.

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External Links:: ISSN 0010-3616, Document, MathReview (Anne Schilling), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

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T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:: §24.4(iii).
Encodings:: BibTeX

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… ►It should be borne in mind that a sequence (series) transformation can be effective for one type of sequence (series) but may not accelerate convergence for another type. It may even fail altogether by not being limit-preserving. … ►Let

k

be a fixed positive integer. … ►The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: … ►For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95). …

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A. L. Van Buren, R. V. Baier, S. Hanish, and B. J. King (1972) Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.

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External Links:: Document, ZentralBlatt
Referenced by:: §30.16(ii), §30.18(iii).
Encodings:: BibTeX

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J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

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B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:: Document
Referenced by:: §14.31(i).
Encodings:: BibTeX

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L. Vietoris (1983) Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 192 (1-3), pp. 83–91 (German).

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External Links:: ISSN 0723-9319, MathReview (Peter Schroth), ZentralBlatt
Referenced by:: §8.10.
Encodings:: BibTeX

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H. Volkmer and J. J. Wood (2014) A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

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External Links:: ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt
Referenced by:: §16.11(ii), §16.11(ii).
Encodings:: BibTeX

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Blanch and Clemm (1965) includes values of ${\operatorname{Mc}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Mc}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=0(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . Also ${\operatorname{Ms}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Ms}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=1(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . In all cases $q=0(.05)1$ . Precision is generally 7D. Approximate formulas and graphs are also included.

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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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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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Blanch and Clemm (1969) includes eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $q=\rho e^{\mathrm{i}\phi}$ , $\rho=0(.5)25$ , $\phi=5^{\circ}(5^{\circ})90^{\circ}$ , $n=0(1)15$ ; 4D. Also $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ for $q=\mathrm{i}\rho$ , $\rho=0(.5)100$ , $n=0(2)14$ and $n=2(2)16$ , respectively; 8D. Double points for $n=0(1)15$ ; 8D. Graphs are included.

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Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n}\left(x,10\right)$ for $n=1(1)10$ , and the first 5 zeros of ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ for $n=0$ or $1(1)8$ , $j=1,2$ . Precision is mostly 9S.

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Bibliography B

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P. Boalch (2005) From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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External Links:: ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

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R. Bulirsch (1965b) Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:: Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.
External Links:: ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt
Referenced by:: §19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).
Encodings:: BibTeX

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P. L. Butzer, M. Hauss, and M. Leclerc (1992) Bernoulli numbers and polynomials of arbitrary complex indices. Appl. Math. Lett. 5 (6), pp. 83–88.

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External Links:: ISSN 0893-9659, MathReview, ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

►

P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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External Links:: ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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N. D. Kazarinoff (1988) Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

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External Links:: ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt
Referenced by:: §16.23(iii).
Encodings:: BibTeX

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R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

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External Links:: ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

T. H. Koornwinder and F. Bouzeffour (2011) Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials. Appl. Anal. 90 (3-4), pp. 731–746.

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External Links:: ISSN 0003-6811, Link, MathReview (Masatoshi Iida), ZentralBlatt
Referenced by:: §18.38(iii), §18.38(iii).
Encodings:: BibTeX

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E. D. Krupnikov and K. S. Kölbig (1997) Some special cases of the generalized hypergeometric function

{}_{q+1}F_{q}

. J. Comput. Appl. Math. 78 (1), pp. 79–95.

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External Links:: ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §16.7.
Encodings:: BibTeX

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M. D. Kruskal (1974) The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.

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External Links:: MathReview (G. L. Lamb, Jr.), ZentralBlatt
Referenced by:: §22.19(iii).
Encodings:: BibTeX

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… ►The following figure illustrates the piece-wise linear mapping of the height to each of the color components red, green and blue, written as

\left\langle R,\;G,\;B\right\rangle

. … ►

\left\langle R,\;G,\;B\right\rangle=\begin{cases}\left\langle 0,\;h,\;1\right% \rangle&\text{if $0\leq h<1$}\\ \left\langle 0,\;1,\;2-h\right\rangle&\text{if $1\leq h<2$}\\ \left\langle h-2,\;1,\;0\right\rangle&\text{if $2\leq h<3$}\\ \left\langle 1,\;4-h,\;0\right\rangle&\text{if $3\leq h\leq 4$}\end{cases}

… ►By painting the surfaces with a color that encodes the phase,

\operatorname{ph}f

, both the magnitude and phase of complex valued functions can be displayed. … ►The conventional CMYK color wheel (not to be confused with the traditional Artist’s color wheel) places the additive colors (red, green, blue) and the subtractive colors (yellow, cyan, magenta) at multiples of 60 degrees. In particular, the colors at 90 and 180 degrees are some vague greenish and purplish hues. …

… ►The same Poincaré expansion holds for

b^{m+1}_{\nu}\left(k^{2}\right)

, since … ►

29.7.7

\tau_{3}=\frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+410p^{2}+405)-24k^{2}(1+k^{2})% ^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+173)),

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Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.8

\tau_{4}=\frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+1260p^{4}+2943p^{2}+486)-8k^{2% }(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16k^{4}(1+k^{2})(35p^{6}+760p^% {4}+2043p^{2}+378)).

ⓘ

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $\tau_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §29.7(i), §29.7 and Ch.29

►Formulas for additional terms can be computed with the author’s Maple program LA5; see §29.22. … ►Müller (1966a, b) found three formal asymptotic expansions for a fundamental system of solutions of (29.2.1) (and (29.11.1)) as

\nu\to\infty

, one in terms of Jacobian elliptic functions and two in terms of Hermite polynomials. …

.%E8%B6%B3%E7%90%83%E4%B8%96%E7%95%8C%E6%9D%2014b%E3%80%8Ewn4.com%E3%80%8F%E4%BB%8A%E5%B9%B4%E4%B8%96%E7%95%8C%E6%9D%AF%E5%BE%AE%E4%BF%A1%E8%B5%8C%E7%90%2022-12-22%E5%B9%411%E6%9C%882%E6%97%523%E6%97%269%E5%88%862%E7%A7%caniuse.com

11—20 of 839 matching pages

11: 25.19 Tables

12: 22.21 Tables

13: Bibliography

14: 3.9 Acceleration of Convergence

15: Bibliography V

16: 28.35 Tables

17: Bibliography B

Bibliography B

18: Bibliography K

19: About Color Map

20: 29.7 Asymptotic Expansions