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11—20 of 674 matching pages

11: 23 Weierstrass Elliptic and Modular
Functions

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12: 19.36 Methods of Computation

… ►For $R_F$ the polynomial of degree 7, for example, is … ►All cases of $R_F$, $R_C$, $R_J$, and $R_D$ are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). … ►The incomplete integrals $R_F(x,y,z)$ and $R_G(x,y,z)$ can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to $R_C$, accompanied by two quadratically convergent series in the case of $R_G$; compare Carlson (1965, §§5,6). … ► $F(\varphi ,k)$ can be evaluated by using (19.25.5). …A summary for $F(\varphi ,k)$ is given in Gautschi (1975, §3). …

13: 3.3 Interpolation

… ►If $f$ is analytic in a simply-connected domain $D$ (§1.13(i)), then for $z\in D$, …where $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing the points $z,z_1,z_2,\mathrm{\dots },z_n$. … ►where ${\omega }_{n+1}(\zeta )$ is given by (3.3.3), and $C$ is a simple closed contour in $D$ described in the positive rotational sense and enclosing $z_0,z_1,\mathrm{\dots },z_n$. … ►By using this approximation to $x$ as a new point, $x_3=x$, and evaluating $\left[f_0,f_1,f_2,f_3\right]x=\mathrm{1.12388\hspace{0.33em}6190}$, we find that $x=-\mathrm{2.33810\hspace{0.33em}7409}$, with 9 correct digits. … ►Then by using $x_3$ in Newton’s interpolation formula, evaluating $\left[x_0,x_1,x_2,x_3\right]f=-\mathrm{0.26608\hspace{0.33em}28233}$ and recomputing $f^{\prime }(x)$, another application of Newton’s rule with starting value $x_3$ gives the approximation $x=\mathrm{2.33810\hspace{0.33em}7373}$, with 8 correct digits. …

14: 14.33 Tables

… ►

Abramowitz and Stegun (1964, Chapter 8) tabulates ${𝖯}_n\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 5–8D; ${𝖯}_n^{\prime }\left(x\right)$ for $n=1(1)4,9,10$, $x=0(.01)1$, 5–7D; ${𝖰}_n\left(x\right)$ and ${𝖰}_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 6–8D; $P_n\left(x\right)$ and $P_n^{\prime }\left(x\right)$ for $n=0(1)5,9,10$, $x=1(.2)10$, 6S; $Q_n\left(x\right)$ and $Q_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=1(.2)10$, 6S. (Here primes denote derivatives with respect to $x$.)

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Zhang and Jin (1996, Chapter 4) tabulates ${𝖯}_n\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; ${𝖯}_n\left(\cos \theta \right)$ for $n=1(1)4,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖰}_n\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; ${𝖰}_n\left(\cos \theta \right)$ for $n=0(1)3,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖯}_n^m\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; ${𝖰}_n^m\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; ${𝖯}_{\nu }^m\left(\cos \theta \right)$ for $m=0(1)3$, $\nu =0(.25)5$, $\theta =0({15}^{\circ }){90}^{\circ }$, 5D; $P_n\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q_n\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of ${𝖯}_{\nu }^m\left(\cos \theta \right)$ and of its derivative for $m=0(1)4$, $\theta ={10}^{\circ },{30}^{\circ },{150}^{\circ }$.

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Belousov (1962) tabulates ${𝖯}_n^m\left(\cos \theta \right)$ (normalized) for $m=0(1)36$, $n-m=0(1)56$, $\theta =0({2.5}^{\circ }){90}^{\circ }$, 6D.

…

15: 18.8 Differential Equations

… ►

Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

► ►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
…
4	$C_n^{(\lambda )}\left(x\right)$	$1-x^2$	$-(2\lambda +1)x$	$0$	$n(n+2\lambda )$
…
8	$L_n^{(\alpha )}\left(x\right)$	$x$	$\alpha +1-x$	$0$	$n$
9	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(x^2\right)$	$1$	$0$	$-x^2+(\frac{1}{4}-{\alpha }^2)x^{-2}$	$4n+2\alpha +2$
…

►

…

16: 15.3 Graphics

… ►

►►

Figure 15.3.1: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x),-100\le x\le 1$. Magnify

►

►►

Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

►

►►

Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

►

►►

Figure 15.3.4: $F(5,10;1;x),-1\le x\le 0.022$. Magnify

… ►

►

►

Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. … Magnify 3D Help

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17: Bibliography T

… ►

The universal Askey-Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$ . SIGMA 9, pp. Paper 047, 40 pp..

ⓘ

External Links:

ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

… ►

Algorithm 926: incomplete gamma functions with negative arguments. ACM Trans. Math. Software 39 (2), pp. Art. 14, 9.

ⓘ

External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

7th item, §8.28(ii).

Encodings:

BibTeX

… ►

Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:

With 1 Macintosh floppy disk (3.5 inch; DD). Programs for the calculation of $3j,6j,9j$ symbols are included.

External Links:

ISBN 0-471-55264-X, MathReview (J. F. Cornwell), ZentralBlatt

Referenced by:

§34.3(vii).

Encodings:

BibTeX

… ►

Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.

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External Links:

Document

Referenced by:

§25.21(vii).

Encodings:

BibTeX

… ►

Theory of the Fresnel integral. USSR Comput. Math. and Math. Phys. 9 (4), pp. 271–279.

ⓘ

Notes:

Translation of the paper in Zh. Vychisl. Mat. Mat. Fiz., Vol. 9, No. 4, 938-944, 1969

External Links:

Document, ZentralBlatt

Referenced by:

§7.13(iv).

Encodings:

BibTeX

18: Bibliography

… ►

Uniform asymptotic expansions for exponential integrals $E_n(x)$ and Bickley functions ${\mathrm{Ki}}_n(x)$ . ACM Trans. Math. Software 9 (4), pp. 467–479.

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External Links:

ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

… ►

Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

(18.27.12_5), (18.27.6).

Encodings:

BibTeX

… ►

Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

BibTeX

… ►

Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

Encodings:

BibTeX

… ►

A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

ⓘ

External Links:

ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§29.11.

Encodings:

BibTeX

…

19: 18.5 Explicit Representations

… ►In (18.5.4_5) see §26.11 for the Fibonacci numbers $F_n$. … ►In this equation $w(x)$ is as in Table 18.3.1, (reproduced in Table 18.5.1), and $F(x)$, ${\kappa }_n$ are as in Table 18.5.1. … ►For the definitions of ${}_2{}^{}F_1^{}$, ${}_1{}^{}F_1^{}$, and ${}_2{}^{}F_0^{}$ see §16.2. … ►

18.5.9 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.46

Referenced by:

§18.5(iii), §18.6(i)

Permalink:

http://dlmf.nist.gov/18.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. …

20: 3.4 Differentiation

… ►The $B_k^n$ are the differentiated Lagrangian interpolation coefficients: …where $A_k^n$ is as in (3.3.10). … ►

$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

… ►where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and $x_0$ is interior to $C$. Taking $C$ to be a circle of radius $r$ centered at $x_0$, we obtain …