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21—30 of 758 matching pages

21: 14.33 Tables

… ►

•

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

►

•

Zhang and Jin (1996, Chapter 4) tabulates $\mathsf{P}_{n}\left(x\right)$ for $n=2(1)5,10$ , $x=0(.1)1$ , 7D; $\mathsf{P}_{n}\left(\cos\theta\right)$ for $n=1(1)4,10$ , $\theta=0(5^{\circ})90^{\circ}$ , 8D; $\mathsf{Q}_{n}\left(x\right)$ for $n=0(1)2,10$ , $x=0(.1)0.9$ , 8S; $\mathsf{Q}_{n}\left(\cos\theta\right)$ for $n=0(1)3,10$ , $\theta=0(5^{\circ})90^{\circ}$ , 8D; $\mathsf{P}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n-m=0(1)2$ , $n=10$ , $x=0,0.5$ , 8S; $\mathsf{Q}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n=0(1)2,10$ , 8S; $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ 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 $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ and of its derivative for $m=0(1)4$ , $\theta=10^{\circ},30^{\circ},150^{\circ}$ .

►

•

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

…

22: Bibliography V

… ►

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.

ⓘ

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

… ►

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.

ⓘ

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

… ►

A. Verma and V. K. Jain (1983) Certain summation formulae for

q

-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

ⓘ

External Links:: ISSN 0019-5839, MathReview Entry, ZentralBlatt
Referenced by:: (17.6.4_5), §17.6(i), (17.8.8), §17.8.
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).

ⓘ

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

… ►

H. Volkmer (2023) Asymptotic expansion of the generalized hypergeometric function

{}_{p}F_{q}(z)

z\to\infty

for

p<q

. Anal. Appl. (Singap.) 21 (2), pp. 535–545.

ⓘ

External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).
Encodings:: BibTeX

…

23: Bibliography S

… ►

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

ⓘ

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

… ►

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.

ⓘ

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

… ►

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

ⓘ

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

… ►

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.

ⓘ

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

… ►

R. Spigler (1980) Some results on the zeros of cylindrical functions and of their derivatives. Rend. Sem. Mat. Univ. Politec. Torino 38 (1), pp. 67–85 (Italian. English summary).

ⓘ

External Links:: MathReview (M. E. Muldoon), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

…

24: Bibliography C

… ►

L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

… ►

B. C. Carlson and J. FitzSimons (2000) Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

ⓘ

Notes:: Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.
External Links:: ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt
Referenced by:: §19.29(ii), §19.36(i), §19.36(i), §19.39(iv).
Encodings:: BibTeX

… ►

M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §8.16.
Encodings:: BibTeX

… ►

W. J. Cody (1965b) Chebyshev polynomial expansions of complete elliptic integrals. Math. Comp. 19 (90), pp. 249–259.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.38.
Encodings:: BibTeX

… ►

J. W. Cooley and J. W. Tukey (1965) An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (90), pp. 297–301.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §3.11(v).
Encodings:: BibTeX

…

25: 29.20 Methods of Computation

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that

n

has to be chosen sufficiently large. … ►Alternatively, the zeros can be found by locating the maximum of function

g

in (29.12.11).

26: 26.12 Plane Partitions

… ►An equivalent definition is that a plane partition is a finite subset of

\mathbb{N}\times\mathbb{N}\times\mathbb{N}

with the property that if

(r,s,t)\in\pi

and

(1,1,1)\leq(h,j,k)\leq(r,s,t)

, then

(h,j,k)

must be an element of

\pi

. …It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point

(h,j,k)\in\pi

. … ►We define the

r\times s\times t

box

B(r,s,t)

as …Then the number of plane partitions in

B(r,s,t)

is … ►The number of symmetric plane partitions in

B(r,r,t)

is …

27: 22.21 Tables

… ►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.

28: Bibliography M

… ►

P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function

{\rm Ai}(x)

. J. Comput. Phys. 99 (2), pp. 337–340.

ⓘ

Notes:: There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$ .
External Links:: ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. W. Matula and P. Kornerup (1980) Foundations of Finite Precision Rational Arithmetic. In Fundamentals of Numerical Computation (Computer-oriented Numerical Analysis), G. Alefeld and R. D. Grigorieff (Eds.), Comput. Suppl., Vol. 2, Vienna, pp. 85–111.

ⓘ

External Links:: MathReview (G. Blanch), ZentralBlatt
Referenced by:: §3.1(iii).
Encodings:: BibTeX

… ►

P. Midy (1975) An improved calculation of the general elliptic integral of the second kind in the neighbourhood of

x=0

. Numer. Math. 25 (1), pp. 99–101.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

A. R. Miller (1997) A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §16.10.
Encodings:: BibTeX

… ►

S. C. Milne (1985c) A new symmetry related to

\mathit{SU}(n)

for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

29: 19.11 Addition Theorems

… ►

19.11.1

F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right),

ⓘ

Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle
Referenced by:: §19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.5

\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left% (\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.6_5

R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right).

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos\NVar{z}$ : cosine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)
Permalink:: http://dlmf.nist.gov/19.11.E6_5
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.28):: This equation was given a number.
See also:: Annotations for §19.11(i), §19.11 and Ch.19

►Hence, care has to be taken with the multivalued functions in (19.11.5). … ►

19.11.7

F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right),

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

…

30: 25.16 Mathematical Applications

… ►In studying the distribution of primes

p\leq x

, Chebyshev (1851) introduced a function

\psi\left(x\right)

(not to be confused with the digamma function used elsewhere in this chapter), given by … ►where

H_{n}

is given by (25.11.33). ►

H\left(s\right)

is analytic for

\Re s>1

, and can be extended meromorphically into the half-plane

\Re s>-2k

for every positive integer

k

by use of the relations … ►For integer

s

(

\geq 2

H\left(s\right)

can be evaluated in terms of the zeta function: … ►

H\left(s\right)

has a simple pole with residue

\zeta\left(1-2r\right)

(

=-B_{2r}/(2r)

) at each odd negative integer

s=1-2r

r=1,2,3,\dots

. …