DLMF: Untitled Document

… ►

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

.

► ►►►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
2	$\begin{gathered}\textstyle\left(\sin\frac{1}{2}x\right)^{\alpha+\frac{1}{2}}% \left(\cos\frac{1}{2}x\right)^{\beta+\frac{1}{2}}\\ \times P^{(\alpha,\beta)}_{n}\left(\cos x\right)\end{gathered}$	$1$	$0$	$\dfrac{\tfrac{1}{4}-\alpha^{2}}{4{\sin}^{2}\frac{1}{2}x}+\dfrac{\frac{1}{4}-% \beta^{2}}{4{\cos}^{2}\frac{1}{2}x}$	$\left(n+\frac{1}{2}(\alpha+\beta+1)\right)^{2}$
…
5	$T_{n}\left(x\right)$	$1-x^{2}$	$-x$	$0$	$n^{2}$
…
9	${\mathrm{e}}^{-\frac{1}{2}x^{2}}x^{\alpha+\frac{1}{2}}L^{(\alpha)}_{n}\left(x^% {2}\right)$	$1$	$0$	$-x^{2}+(\frac{1}{4}-\alpha^{2})x^{-2}$	$4n+2\alpha+2$
…
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
13	${\mathrm{e}}^{-\frac{1}{2}x^{2}}H_{n}\left(x\right)$	$1$	$0$	$-x^{2}$	$2n+1$
…

► …

… ►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)

. ►Curtis (1964b) tabulates

\operatorname{sn}\left(mK/n,k\right)

,

\operatorname{cn}\left(mK/n,k\right)

,

\operatorname{dn}\left(mK/n,k\right)

for

n=2(1)15

,

m=1(1)n-1

, and

q

(not

k

)

=0(.005)0.35

to 20D. … ►5 to 2. 2. … ►Zhang and Jin (1996, p. 678) tabulates

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

,

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

,

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

for

k=\frac{1}{4},\frac{1}{2}

and

x=0(.1)4

to 7D. …

… ►

•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

… ►

•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

… ►

•

Jahnke and Emde (1945) tabulates $\mathbf{E}_{n}\left(x\right)$ for $n=1,2$ and $x=0(.01)14.99$ to 4D.

… ►

•

Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function $\mathbf{H}_{n}\left(x,\alpha\right)$ for $n=0,1$ , $x=0(.2)10$ , and $\alpha=0(.2)1.4,\tfrac{1}{2}\pi$ , together with surface plots.

… ► ►►

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

… ►

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},

►

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

►

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}.

►An often used alternative to the

\mathit{3j}

symbol is the Clebsch–Gordan coefficient …

… ►For example,

p\left(5\right)=7

because there are exactly seven partitions of

5

:

5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1

. … ►where the exponents

1

,

2

,

5

,

7

,

12

,

15

,

\dots

are the pentagonal numbers, defined by … ►where

K=\pi\sqrt{2/3}

(Hardy and Ramanujan (1918)). … ►where

\varepsilon=\exp\left(\pi\mathrm{i}(((a+d)/(12c))-s(d,c))\right)

and

s(d,c)

is given by (27.14.11). … ►The 24th power of

\eta\left(\tau\right)

in (27.14.12) with

e^{2\pi\mathrm{i}\tau}=x

is an infinite product that generates a power series in

x

with integer coefficients called Ramanujan’s tau function

\tau\left(n\right)

: …

… ►

28.26.1

{\operatorname{Mc}^{(3)}_{m}}\left(z,h\right)=\dfrac{e^{\mathrm{i}\phi}}{(\pi h% \cosh z)^{\ifrac{1}{2}}}\*\left(\mathrm{Fc}_{m}\left(z,h\right)-\mathrm{i}% \mathrm{Gc}_{m}\left(z,h\right)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $\phi$
A&S Ref:: 20.9.7 (in slightly different notation)
Permalink:: http://dlmf.nist.gov/28.26.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

… ►

28.26.3

\phi=2h\sinh z-\left(m+\tfrac{1}{2}\right)\operatorname{arctan}\left(\sinh z% \right).

ⓘ

Symbols:: $\sinh\NVar{z}$ : hyperbolic sine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $\phi$
A&S Ref:: 20.9.8 (in slightly different notation)
Permalink:: http://dlmf.nist.gov/28.26.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

►Then as

h\to+\infty

with fixed

z

in

\Re z>0

and fixed

s=2m+1

, ►

28.26.4

\mathrm{Fc}_{m}\left(z,h\right)\sim 1+\dfrac{s}{8h{\cosh}^{2}z}+\dfrac{1}{2^{1% 1}h^{2}}\left(\dfrac{s^{4}+86s^{2}+105}{{\cosh}^{4}z}-\dfrac{s^{4}+22s^{2}+57}% {{\cosh}^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(-\dfrac{s^{5}+14s^{3}+33s}{{% \cosh}^{2}z}-\dfrac{2s^{5}+124s^{3}+1122s}{{\cosh}^{4}z}+\dfrac{3s^{5}+290s^{3% }+1627s}{{\cosh}^{6}z}\right)+\cdots,

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cosh\NVar{z}$ : hyperbolic cosine function, $m$ : integer, $h$ : parameter and $z$ : complex variable
A&S Ref:: 20.9.9 (in slightly different notation)
Referenced by:: §28.26(i)
Permalink:: http://dlmf.nist.gov/28.26.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.5

\mathrm{Gc}_{m}\left(z,h\right)\sim\dfrac{\sinh z}{{\cosh}^{2}z}\left(\dfrac{s% ^{2}+3}{2^{5}h}+\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cosh}% ^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(5s^{4}+34s^{2}+9-\dfrac{s^{6}-47s^{4% }+667s^{2}+2835}{12{\cosh}^{2}z}+\dfrac{s^{6}+505s^{4}+12139s^{2}+10395}{12{% \cosh}^{4}z}\right)\right)+\cdots.

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $m$ : integer, $h$ : parameter and $z$ : complex variable
A&S Ref:: 20.9.10 (in slightly different notation)
Referenced by:: §28.26(i)
Permalink:: http://dlmf.nist.gov/28.26.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.26(i), §28.26 and Ch.28

…

… ►

M. K. Kerimov and S. L. Skorokhodov (1985c) Calculation of the multiple zeros of the derivatives of the cylindrical Bessel functions

J_{\nu}(z)

and

Y_{\nu}(z)

. Zh. Vychisl. Mat. i Mat. Fiz. 25 (12), pp. 1749–1760, 1918.

ⓘ

Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 25(6), pp. 101–107
External Links:: ISSN 0044-4669, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §10.74(vi), 10th item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1999) The Rayleigh function: Theory and computational methods. Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.

ⓘ

Notes:: Translation in Comput. Math. Math. Phys. 39 (1999), No. 12, pp. 1883–1925.
External Links:: ISSN 0044-4669, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.21(xiii).
Encodings:: BibTeX

… ►

S. F. Khwaja and A. B. Olde Daalhuis (2013) Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2153), pp. 20130008, 12.

ⓘ

External Links:: ISSN 1364-5021, Document, FullText, MathReview (Ulrich D. Jentschura), ZentralBlatt
Referenced by:: §2.4(vi), §2.4(vi).
Encodings:: BibTeX

… ►

G. C. Kokkorakis and J. A. Roumeliotis (1998) Electromagnetic eigenfrequencies in a spheroidal cavity (calculation by spheroidal eigenvectors). J. Electromagn. Waves Appl. 12 (12), pp. 1601–1624.

ⓘ

External Links:: ISSN 0920-5071, Document, MathReview, ZentralBlatt
Referenced by:: §30.14(v).
Encodings:: BibTeX

… ►

S. Kowalevski (1889) Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1), pp. 177–232 (French).

ⓘ

External Links:: ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt
Referenced by:: §22.19(iv).
Encodings:: BibTeX

…

… ►Tables of zonal polynomials are given in James (1964) for

|\kappa|\leq 6

, Parkhurst and James (1974) for

|\kappa|\leq 12

, and Muirhead (1982, p. 238) for

|\kappa|\leq 5

. …

… ►

L. V. Babushkina, M. K. Kerimov, and A. I. Nikitin (1997) New tables of Bessel functions of complex argument. Comput. Math. Math. Phys. 37 (12), pp. 1480–1482.

ⓘ

Notes:: Russian original in Zh. Vychisl. Mat. i Mat. Fiz. 37(1997), no. 12, 1526–1528
External Links:: ZentralBlatt
Referenced by:: §10.75(i).
Encodings:: BibTeX

… ►

J. Baik, P. Deift, and K. Johansson (1999) On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (4), pp. 1119–1178.

ⓘ

External Links:: ISSN 0894-0347, FullText, MathReview (David J. Aldous), ZentralBlatt
Referenced by:: §32.14.
Encodings:: BibTeX

… ►

W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

ⓘ

External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

… ►

M. V. Berry (1977) Focusing and twinkling: Critical exponents from catastrophes in non-Gaussian random short waves. J. Phys. A 10 (12), pp. 2061–2081.

ⓘ

External Links:: Document, MathReview, ZentralBlatt
Referenced by:: §36.6.
Encodings:: BibTeX

… ►

J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, and M. A. Shokrollahi (2001) Irregular primes and cyclotomic invariants to 12 million. J. Symbolic Comput. 31 (1-2), pp. 89–96.

ⓘ

Notes:: Computational algebra and number theory (Milwaukee, WI, 1996)
External Links:: ISSN 0747-7171, Document, MathReview (Xavier-François Roblot), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

…

.2022年足球世界杯冠军_『网址:68707.vip』足球比赛赌用什么app_b5p6v3_2022年12月2日6时27分38秒_3xvvl7r9d.cc

31—40 of 810 matching pages

31: 18.8 Differential Equations

32: 22.21 Tables

33: 11.14 Tables

34: 34.1 Special Notation

35: 27.14 Unrestricted Partitions

36: 28.26 Asymptotic Approximations for Large $q$

37: Bibliography K

38: 11 Struve and Related Functions

39: 35.11 Tables

40: Bibliography B