DLMF: Untitled Document

… ►

G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

ⓘ

External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

… ►

E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near

T_{c}

. Phys. Rev. Lett. 31, pp. 1409–1411.

ⓘ

External Links:: Document
Referenced by:: §32.16.
Encodings:: BibTeX

… ►

K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

… ►

W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

… ►

P. Boalch (2006) The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, pp. 183–214.

ⓘ

External Links:: ISSN 0075-4102, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

…

… ►For

z=0

, the set consists of the two curves …where

x_{\pm}

are the two smallest positive roots of the equation … ►For

z>0

the Stokes set has two sheets. … ►This part of the Stokes set connects two complex saddles. … ►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …

… ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►The first two coefficients are given by ►

12.11.5

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

ⓘ

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

… ►

12.11.8

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

ⓘ

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

… ►

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),

ⓘ

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

…

… ►

V. Laĭ (1994) The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).

ⓘ

Notes:: In Russian. English translation: Theoret. and Math. Phys. 101(1994), no. 3, pp. 1413–1418 (1995)
External Links:: ISSN 0564-6162, Document, MathReview, ZentralBlatt
Referenced by:: §31.18.
Encodings:: BibTeX

… ►

W. Lay and S. Yu. Slavyanov (1998) The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

… ►

E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt
Referenced by:: §30.12, §31.17(ii).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

E. R. Love (1972b) Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.

ⓘ

External Links:: Document, MathReview (C. Fox), ZentralBlatt
Referenced by:: §1.15(vi), §1.15(vii).
Encodings:: BibTeX

…

… ►

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),

ⓘ

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

…

… ►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

.

ⓘ

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}.

ⓘ

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

…

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►

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.

ⓘ

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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

…

… ►There are many ways to implement these first two steps, noting that the expressions for

\alpha_{n}

and

\beta_{n}

of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010). … ►

18.40.5

F_{N}(z)=\frac{1}{\mu_{0}}\sum_{n=1}^{N}\frac{w_{n}}{z-x_{n}}.

ⓘ

Symbols:: $N$ : positive integer, $w_{x}$ : weights, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.40.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►Results of low (

~{}2

to

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

to

20

. … ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: ►

18.40.9

x(t,N)=\cfrac{x_{1,N}}{1+\cfrac{a_{1}(t-1)}{1+\cfrac{a_{2}(t-2)}{1+\cdots}}}% \frac{a_{N-1}(t-(N-1))}{1},

t\in(0,\infty)

,

ⓘ

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $N$ : positive integer, $t$ : real variable and $x$ : real variable
Source:: Schlessinger (1968)
Permalink:: http://dlmf.nist.gov/18.40.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

…

… ►

§22.3(i) Real Variables: Line Graphs

… ►

§22.3(ii) Real Variables: Surfaces

… ►

See accompanying text — Figure 22.3.13: $\operatorname{sn}\left(x,k\right)$ for $k=1-e^{-n}$ , $n=0$ to 20, $-5\pi\leq x\leq 5\pi$ . Magnify 3D Help

… ►

§22.3(iii) Complex $z$ ; Real $k$

… ►

…

… ►

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail, J. Letessier, G. Valent, and J. Wimp (1990) Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.

ⓘ

External Links:: ISSN 0008-414X, MathReview (Mizan Rahman), ZentralBlatt
Referenced by:: §18.26(iv).
Encodings:: BibTeX

… ►

M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

ⓘ

External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt
Referenced by:: §1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

►

M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

Notes:: With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.
External Links:: ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt
Referenced by:: §18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

…

two%20variables

1—10 of 15 matching pages

1: Bibliography B

2: 36.5 Stokes Sets

3: 12.11 Zeros

§12.11(ii) Asymptotic Expansions of Large Zeros

4: Bibliography L

5: 23.9 Laurent and Other Power Series

6: 26.14 Permutations: Order Notation

7: Bibliography K

8: 18.40 Methods of Computation

9: 22.3 Graphics

§22.3(i) Real Variables: Line Graphs

§22.3(ii) Real Variables: Surfaces

§22.3(iii) Complex $z$ ; Real $k$

10: Bibliography I

two%20variables

1—10 of 15 matching pages

1: Bibliography B

2: 36.5 Stokes Sets

3: 12.11 Zeros

§12.11(ii) Asymptotic Expansions of Large Zeros

4: Bibliography L

5: 23.9 Laurent and Other Power Series

6: 26.14 Permutations: Order Notation

7: Bibliography K

8: 18.40 Methods of Computation

9: 22.3 Graphics

§22.3(i) Real Variables: Line Graphs

§22.3(ii) Real Variables: Surfaces

§22.3(iii) Complex z ; Real k

10: Bibliography I

§22.3(iii) Complex $z$ ; Real $k$