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21—30 of 620 matching pages

21: 10.9 Integral Representations

… ►

10.9.30

{J_{\nu}}^{2}\left(z\right)+{Y_{\nu}}^{2}\left(z\right)=\frac{8}{\pi^{2}}\int_% {0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2z\sinh t\right)\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{2}\pi

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

…

22: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►General references for this subsection include Friedman (1990, pp. 4–6), Shilov (2013, pp. 249–256), Riesz and Sz.-Nagy (1990, Ch. 5, §82). … ►For

f(x)\in C(X)\cap L^{2}\left(X\right)\cap\mathcal{D}(T)

f(x)

has the eigenfunction expansion, following directly from (1.18.17)–(1.18.19), … ►The implicit boundary conditions taken here are that the

\phi_{n}(x)

and

\phi_{n}^{\prime}(x)

vanish as

x\to\pm\infty

, which in this case is equivalent to requiring

\phi_{n}(x)\in L^{2}\left(X\right)

, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point. … ►More generally, for

f\in C(X)

x\in X

, see (1.4.24), … ►More generally, continuous spectra may occur in sets of disjoint finite intervals

[\lambda_{a},\lambda_{b}]\in(0,\infty)

, often called bands, when

q(x)

is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). …

23: 12.14 The Function $W\left(a,x\right)$

… ►For the modulus functions

\widetilde{F}(a,x)

and

\widetilde{G}(a,x)

see §12.14(x). … ►Other expansions, involving

\cos\left(\tfrac{1}{4}x^{2}\right)

and

\sin\left(\tfrac{1}{4}x^{2}\right)

, can be obtained from (12.4.3) to (12.4.6) by replacing

a

-ia

and

z

xe^{\ifrac{\pi i}{4}}

; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►Here

{\cal{A}}_{s}(t)

is as in §12.10(ii),

\sigma

is defined by … ►uniformly for

t\in[-1+\delta,1-\delta]

, with

\eta

given by (12.10.23) and

{\widetilde{\cal A}}_{s}(t)

given by (12.10.24). … ►uniformly for

t\in[-1+\delta,\infty)

, with

\zeta

\phi(\zeta)

A_{s}(\zeta)

, and

B_{s}(\zeta)

as in §12.10(vii). …

24: Bibliography B

… ►

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

… ►

A. P. Bassom, P. A. Clarkson, and A. C. Hicks (1995) Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.10(iv), §32.2(iii), §32.7(iv), §32.8(iv).
Encodings:: BibTeX

… ►

D. K. Bhaumik and S. K. Sarkar (2002) On the power function of the likelihood ratio test for MANOVA. J. Multivariate Anal. 82 (2), pp. 416–421.

ⓘ

External Links:: ISSN 0047-259X, Document, MathReview (Yasuko Chikuse), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

R. Bo and R. Wong (1996) Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt
Referenced by:: §18.35(iii).
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

…

25: 1.6 Vectors and Vector-Valued Functions

… ►The geometrical image

C

of a path

\mathbf{c}

is called a simple closed curve if

\mathbf{c}

is one-to-one, with the exception

\mathbf{c}(a)=\mathbf{c}(b)

. The curve

C

is piecewise differentiable if

\mathbf{c}

is piecewise differentiable. Note that

C

can be given an orientation by means of

\mathbf{c}

. … ►and

S

be the closed and bounded point set in the

(x,y)

plane having a simple closed curve

C

as boundary. If

C

is oriented in the positive (anticlockwise) sense, then …

… ►For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).

28: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

29: 19.2 Definitions

… ►The integral for

E\left(\phi,k\right)

is well defined if

k^{2}={\sin}^{2}\phi=1

, and the Cauchy principal value (§1.4(v)) of

\Pi\left(\phi,\alpha^{2},k\right)

is taken if

1-\alpha^{2}{\sin}^{2}\phi

vanishes at an interior point of the integration path. … ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

… ►Formulas involving

\Pi\left(\phi,\alpha^{2},k\right)

that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using

R_{C}\left(x,y\right)

. … ►When

x

and

y

are positive,

R_{C}\left(x,y\right)

is an inverse circular function if

x<y

and an inverse hyperbolic function (or logarithm) if

x>y

: …For the special cases of

R_{C}\left(x,x\right)

and

R_{C}\left(0,y\right)

see (19.6.15). …

30: Errata

… ►

Paragraph Prime Number Theorem (in §27.12)

The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

… ►

Version 1.0.11 (June 8, 2016)

… ►

Equation (10.19.11)

10.19.11

Q_{3}\left(a\right)=\tfrac{549}{28000}a^{8}-\tfrac{1\;10767}{6\;93000}a^{5}+% \tfrac{79}{12375}a^{2}

Originally the first term on the right-hand side of this equation was written incorrectly as $-\tfrac{549}{28000}a^{8}$ .

Reported 2015-03-16 by Svante Janson.

… ►

Table 26.8.1

Originally the Stirling number $s\left(10,6\right)$ was given incorrectly as 6327. The correct number is 63273.

$n$	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
10	$0$	$-3\,62880$	$10\,26576$	$-11\,72700$	$7\,23680$	$-2\,69325$	$63273$	$-9450$	$870$	$-45$	$1$

Reported 2013-11-25 by Svante Janson.

… ►

Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

…

21—30 of 620 matching pages

21: 10.9 Integral Representations

22: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

23: 12.14 The Function $W\left(a,x\right)$

24: Bibliography B

25: 1.6 Vectors and Vector-Valued Functions

26: 10.21 Zeros

27: 14.17 Integrals

28: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

29: 19.2 Definitions

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

30: Errata

Version 1.0.11 (June 8, 2016)

21—30 of 620 matching pages

21: 10.9 Integral Representations

22: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

23: 12.14 The Function W ⁡ ( a , x )

24: Bibliography B

25: 1.6 Vectors and Vector-Valued Functions

26: 10.21 Zeros

27: 14.17 Integrals

28: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

29: 19.2 Definitions

§19.2(iv) A Related Function: R C ⁡ ( x , y )

30: Errata

Version 1.0.11 (June 8, 2016)

23: 12.14 The Function $W\left(a,x\right)$

28: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$