DLMF: Untitled Document

… ►

C. W. Schelin (1983) Calculator function approximation. Amer. Math. Monthly 90 (5), pp. 317–325.

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External Links:: ISSN 0002-9890, FullText, MathReview (J. Albrycht), ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

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B. I. Schneider, J. Segura, A. Gil, X. Guan, and K. Bartschat (2010) A new Fortran 90 program to compute regular and irregular associated Legendre functions. Comput. Phys. Comm. 181 (12), pp. 2091–2097.

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External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 8th item, §14.34(ii).
Encodings:: BibTeX

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L. Shen (1981) The elliptical microstrip antenna with circular polarization. IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.

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External Links:: FullText
Referenced by:: 5th item.
Encodings:: BibTeX

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R. Sips (1959) Représentation asymptotique des fonctions de Mathieu et des fonctions sphéroidales. II. Trans. Amer. Math. Soc. 90 (2), pp. 340–368.

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External Links:: FullText, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §28.8(ii).
Encodings:: BibTeX

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D. M. Smith (2001) Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions. ACM Trans. Math. Software 27 (4), pp. 377–387.

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External Links:: ISSN 0098-3500, Document, GAMS (module 814; package TOMS), ZentralBlatt
Referenced by:: 2nd item, 6th item, 3rd item, 1st item, 6th item, 5th item.
Encodings:: BibTeX

…

… ►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)

. … ►Lawden (1989, pp. 280–284 and 293–297) tabulates

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

,

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

,

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

,

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

,

\mathrm{Z}\left(x|k\right)

to 5D for

k=0.1(.1)0.9

,

x=0(.1)X

, where

X

ranges from 1. …

… ►

•

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).

… ►

•

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.

…

… ►

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

… ►

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

…

… ►

3.9.5

\ln 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots=\frac{1}{1\cdot 2^{1}}+% \frac{1}{2\cdot 2^{2}}+\frac{1}{3\cdot 2^{3}}+\cdots,

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function
Permalink:: http://dlmf.nist.gov/3.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(ii), §3.9 and Ch.3

►

3.9.6

\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{1}{2}\left(1+% \frac{1!}{1\cdot 3}+\frac{2!}{3\cdot 5}+\frac{3!}{3\cdot 5\cdot 7}+\cdots% \right).

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/3.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(ii), §3.9 and Ch.3

… ►For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95). … ►

Table 3.9.1: Shanks’ transformation for

s_{n}=\sum_{j=1}^{n}(-1)^{j+1}j^{-2}

.

► ►►►►

$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
…
3	0.82300 13550 14	0.82247 78118 35	0.82246 72851 83	0.82246 70397 56	0.82246 70335 90
…
8	0.82243 73137 33	0.82246 67719 32	0.82246 70301 49	0.82246 70333 73	0.82246 70334 23
…

► …

… ►

G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

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A. L. Van Buren and J. E. Boisvert (2002) Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.

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Notes:: For software, see Van Buren (website)
External Links:: ISSN 0033-569X, MathReview (Alan M. Cohen), ZentralBlatt
Referenced by:: §30.16(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

… ►

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

…

… ►

•

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.

… ►

•

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: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

… ►

•

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.

… ►

•

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.

… ►

•

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.

…

… ►

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

… ►

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

… ►

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

… ►

J. G. Byatt-Smith (2000) The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:: ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

…

… ►

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

… ►

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

… ►

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.

ⓘ

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

… ►

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.

ⓘ

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

…

… ►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}

… ►In particular, the colors at 90 and 180 degrees are some vague greenish and purplish hues. ►We therefore use a piecewise linear mapping as illustrated below, that takes phase

0

to red,

\pi/2

to yellow,

\pi

to cyan and

3\pi/2

to blue. ►

…

Figure 3: Continuous phase mapping

…

.2022%E5%B9%B4%E8%B6%B3%E7%90%83%E4%B8%96%E7%95%8C%E6%9D%AF%E5%86%A0%E5%86%9B_%E3%80%8E%E7%BD%91%E5%9D%80%3A68707.vip%E3%80%8F%E8%B6%B3%E7%90%83%E6%AF%94%E8%B5%9B%E8%B5%8C%E7%94%A8%E4%BB%80%E4%B9%88app_b5p6v3_2022%E5%B9%B412%E6%9C%882%E6%97%A56%E6%97%B627%E5%88%8638%E7%A7%92_3xvvl7r9d.cc

11—20 of 743 matching pages

11: Bibliography S

12: 22.21 Tables

13: 25.19 Tables

14: Bibliography

15: 3.9 Acceleration of Convergence

16: Bibliography V

17: 28.35 Tables

18: Bibliography B

19: Bibliography K

20: About Color Map