two%20or%20more%20variables

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11: Bibliography K

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D. E. Knuth (1992) Two notes on notation. Amer. Math. Monthly 99 (5), pp. 403–422.

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External Links:: ISSN 0002-9890, FullText, MathReview (W. Dörfler), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations, Notations.
Encodings:: BibTeX

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K. S. Kölbig (1972a) Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §8.22(ii).
Encodings:: BibTeX

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T. H. Koornwinder (1975c) Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.

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External Links:: MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.37(ii).
Encodings:: BibTeX

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Y. A. Kravtsov (1968) Two new asymptotic methods in the theory of wave propagation in inhomogeneous media. Sov. Phys. Acoust. 14, pp. 1–17.

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External Links:: ISSN 0038-562X
Referenced by:: §36.14(iv).
Encodings:: BibTeX

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V. B. Kuznetsov (1992) Equivalence of two graphical calculi. J. Phys. A 25 (22), pp. 6005–6026.

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External Links:: ISSN 0305-4470, Document, MathReview (Niky Kamran), ZentralBlatt
Referenced by:: §31.17(i).
Encodings:: BibTeX

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12: Bibliography S

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H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:: ZentralBlatt
Referenced by:: §12.13(i).
Encodings:: BibTeX

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H. Shanker (1940c) On the expansion of the product of two parabolic cylinder functions of non integral order. Proc. Benares Math. Soc. (N. S.) 2, pp. 61–68.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

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G. Shanmugam (1978) Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model. In Proceedings of the Conference on Mathematical Analysis and its Applications (Inst. Engrs., Mysore, 1977), Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.

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External Links:: MathReview
Referenced by:: §12.17.
Encodings:: BibTeX

►

J. Shao and P. Hänggi (1998) Decoherent dynamics of a two-level system coupled to a sea of spins. Phys. Rev. Lett. 81 (26), pp. 5710–5713.

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External Links:: Document
Referenced by:: §11.12.
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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13: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form

e^{\pm in\phi}e^{\pm\kappa z}J_{n}\left(\kappa r\right)

, from which a solution satisfying prescribed boundary conditions may be constructed. … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging. …On separation of variables into cylindrical coordinates, the Bessel functions

J_{n}\left(x\right)

, and modified Bessel functions

I_{n}\left(x\right)

and

K_{n}\left(x\right)

, all appear. … ►The functions

\mathsf{j}_{n}\left(x\right)

\mathsf{y}_{n}\left(x\right)

{\mathsf{h}^{(1)}_{n}}\left(x\right)

, and

{\mathsf{h}^{(2)}_{n}}\left(x\right)

arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates

\rho,\theta,\phi

(§1.5(ii)): …

14: 26.14 Permutations: Order Notation

… ►The permutation

35247816

has two descents:

52

and

81

. … ►

26.14.4

\sum_{n,k=0}^{\infty}\genfrac{<}{>}{0.0pt}{}{n}{k}x^{k}\,\frac{t^{n}}{n!}=% \frac{1-x}{\exp((x-1)t)-x},

|x|<1,|t|<1

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Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $\exp\NVar{z}$ : exponential function, $!$ : factorial (as in $n!$ ), $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: §26.14(iii)
Permalink:: http://dlmf.nist.gov/26.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(ii), §26.14 and Ch.26

►

26.14.5

\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{x+k}{n}=x% ^{n}.

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Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(ii), §26.14 and Ch.26

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15: 23.9 Laurent and Other Power Series

… ►

c_{2}=\frac{1}{20}g_{2},

… ►

23.9.6

\wp\left(\omega_{j}+t\right)=e_{j}+(3e_{j}^{2}-5c_{2})t^{2}+(10c_{2}e_{j}+21c_% {3})t^{4}+(7c_{2}e_{j}^{2}+21c_{3}e_{j}+5c_{2}^{2})t^{6}+O\left(t^{8}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $c_{n}$ and $t$ : variable
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

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16: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … ►In the Appendix of the last reference it is shown how to compute

R_{J}

without computing

R_{C}

more than once. … ►The incomplete integrals

R_{F}\left(x,y,z\right)

and

R_{G}\left(x,y,z\right)

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). … ►Bulirsch (1969a, b) extend Bartky’s transformation to

\operatorname{el3}\left(x,k_{c},p\right)

by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

17: Bibliography D

… ►

C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire

Mx+N

. Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:: Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.
External Links:: MathReview, ZentralBlatt
Referenced by:: §25.10(i), §27.11, §27.2(i).
Encodings:: BibTeX

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B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, and M. Schmies (2004) Computing Riemann theta functions. Math. Comp. 73 (247), pp. 1417–1442.

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Notes:: Two of the authors (Deconinck and van Hoeij) have developed Maple code for the algorithms described in this paper.
External Links:: ISSN 0025-5718, Document, Code, MathReview, ZentralBlatt
Referenced by:: §21.10(i), 1st item, §21.2(i), §21.4, M. Heil (1995).
Encodings:: BibTeX

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B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:: ZentralBlatt
Referenced by:: §15.4(ii).
Encodings:: BibTeX

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T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:: (This reference has several typographical errors.)
External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.21(ii), §13.21(iii), §13.21(iv).
Encodings:: BibTeX

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18: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

… ►The first two coefficients are given by ►

12.11.5

p_{0}(\zeta)=t(\zeta),

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Symbols:: $\zeta$ : change of variable and $p_{n}(\zeta)$ : coefficients
Permalink:: http://dlmf.nist.gov/12.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.8

q_{0}(\zeta)=t(\zeta).

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Symbols:: $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/12.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.9

u_{a,1}\sim 2^{\frac{1}{2}}\mu\left(1-1.85575\;708\mu^{-4/3}-0.34438\;34\mu^{-% 8/3}-0.16871\;5\mu^{-4}-0.11414\mu^{-16/3}-0.0808\mu^{-20/3}-\cdots\right),

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $a$ : real or complex parameter and $u_{a,s}$ : zeros
Referenced by:: §12.11(iii)
Permalink:: http://dlmf.nist.gov/12.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §12.11(iii), §12.11 and Ch.12

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19: 11.6 Asymptotic Expansions

… ►

11.6.1

\mathbf{K}_{\nu}\left(z\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{\Gamma% \left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\Gamma\left(\nu+\tfrac{% 1}{2}-k\right)},

|\operatorname{ph}z|\leq\pi-\delta

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.1.29
Referenced by:: §11.6(i), §11.6(i), §11.6(i), §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

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11.6.2

\mathbf{M}_{\nu}\left(z\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}(-1)^{k+1}% \frac{\Gamma\left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\Gamma\left% (\nu+\tfrac{1}{2}-k\right)},

|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.2.6
Referenced by:: §11.6(i), §11.6(i), §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

… ►

11.6.5

\mathbf{H}_{\nu}\left(z\right),\mathbf{L}_{\nu}\left(z\right)\sim\frac{z}{\pi% \nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},

|\operatorname{ph}\nu|\leq\pi-\delta

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Symbols:: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order and $\delta$ : arbitrary small positive constant
Referenced by:: (11.11.7)
Permalink:: http://dlmf.nist.gov/11.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(ii), §11.6 and Ch.11

►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … ►

c_{3}(\lambda)=20\lambda^{-6}-4\lambda^{-4},

…

20: 26.10 Integer Partitions: Other Restrictions

… ►If more than one restriction applies, then the restrictions are separated by commas, for example,

p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)

. … ►

Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from

A_{j,k}

► ►►►

	$p\left(\mathcal{D},n\right)$	$p\left(\mathcal{D}2,n\right)$	$p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)$	$p\left(\mathcal{D}^{\prime}3,n\right)$
…
$20$	$64$	$31$	$20$	$18$

► … ►

26.10.3

(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,\mathcal{D},n\right)x^{m}q^{n}=% \sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}q^{m(m+1)/2}x^{m}=\prod_{j=1}^{% k}(1+x\,q^{j}),

|x|<1

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Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)$ : number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $q$ : parameter
Referenced by:: §26.10(ii)
Permalink:: http://dlmf.nist.gov/26.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.10(ii), §26.10 and Ch.26

… ►

26.10.20

[\![x]\!]=\begin{cases}x-\left\lfloor x\right\rfloor-\tfrac{1}{2},&x\notin% \mathbb{Z},\\ 0,&x\in\mathbb{Z}.\end{cases}

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Symbols:: $\in$ : element of, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\mathbb{Z}$ : set of all integers, $\notin$ : not an element of and $x$ : real variable
Permalink:: http://dlmf.nist.gov/26.10.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §26.10(vi), §26.10 and Ch.26

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