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12: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The minor

M_{jk}

of the entry

a_{jk}

in the

n

th-order determinant

\det[a_{jk}]

is the (

n-1

)th-order determinant derived from

\det[a_{jk}]

by deleting the

j

th row and the

k

th column. … ►for every distinct pair of

j, k

, or when one of the factors

\sum^{n}_{k=1}a^{2}_{jk}

vanishes. … ►where

\omega_{1},\omega_{2},\dots,\omega_{n}

are the

n

th roots of unity (1.11.21). … ►Let

a_{j,k}

be defined for all integer values of

j

and

k

, and

D_{n}[a_{j,k}]

denote the

(2n+1)\times(2n+1)

determinant … ►The corresponding eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

can be chosen such that they form a complete orthonormal basis in

\mathbf{E}_{n}

. …

13: 14.26 Uniform Asymptotic Expansions

… ►The uniform asymptotic approximations given in §14.15 for

P^{-\mu}_{\nu}\left(x\right)

and

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)

for

1<x<\infty

are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …

14: 3.9 Acceleration of Convergence

… ►A transformation of a convergent sequence

\{s_{n}\}

with limit

\sigma

into a sequence

\{t_{n}\}

is called limit-preserving if

\{t_{n}\}

converges to the same limit

\sigma

. … ►If

S=\sum_{k=0}^{\infty}(-1)^{k}a_{k}

is a convergent series, then … ►with

s_{\infty}=\frac{1}{12}{\pi}^{2}=0.82246\;70334\;24\dotsc

. … ►We give a special form of Levin’s transformation in which the sequence

s=\{s_{n}\}

of partial sums

s_{n}=\sum_{j=0}^{n}a_{j}

is transformed into: … ►where

{\left(a\right)_{0}}=1

and

{\left(a\right)_{j}}=a(a+1)\cdots(a+j-1)

are Pochhammer symbols (§5.2(iii)), and the constants

\beta

and

\gamma

are chosen arbitrarily subject to certain conditions. …

15: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►These are given by the following equations in which

a_{1},a_{2},\ldots,a_{n}

are nonnegative integers such that …

M_{1}

is the multinominal coefficient (26.4.2): …

M_{2}

is the number of permutations of

\{1,2,\ldots,n\}

with

a_{1}

cycles of length 1,

a_{2}

cycles of length 2,

\dotsc

, and

a_{n}

cycles of length

n

: …For each

n

all possible values of

a_{1},a_{2},\ldots,a_{n}

are covered. … ►where the summation is over all nonnegative integers

n_{1},n_{2},\ldots,n_{k}

such that

n_{1}+n_{2}+\cdots+n_{k}=n

. …

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.

►

•

Agrest et al. (1982) tabulates $\mathbf{H}_{n}\left(x\right)$ and $e^{-x}\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

… ►

•

Zanovello (1975) tabulates $\mathbf{H}_{n}\left(x\right)$ for $n=-4(1)15$ and $x=0.5(.5)26$ to 8D or 9S.

►

•

Zhang and Jin (1996) tabulates $\mathbf{H}_{n}\left(x\right)$ and $\mathbf{L}_{n}\left(x\right)$ for $n=-4(1)3$ and $x=0(1)20$ to 8D or 7S.

… ►

•

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.

…

18: 19.29 Reduction of General Elliptic Integrals

… ►It can be expressed in terms of symmetric integrals by setting

a_{5}=1

and

b_{5}=0

in (19.29.8). … ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as

R_{G}\left(a_{1}b_{2},a_{2}b_{1},0\right)

multiplied either by

-2/(b_{1}b_{2})

or by

-2/(a_{1}a_{2})

in the cases of (19.29.20) or (19.29.21), respectively. … ►Next, for

j=1,2

, define

Q_{j}(t)=f_{j}+g_{j}t+h_{j}t^{2}

, and assume both

Q

’s are positive for

y<t<x

. …If

Q_{1}(t)=(a_{1}+b_{1}t)(a_{2}+b_{2}t)

, where both linear factors are positive for

y<t<x

, and

Q_{2}(t)=f_{2}+g_{2}t+h_{2}t^{2}

, then (19.29.25) is modified so that …In the cubic case, in which

a_{2}=1

b_{2}=0

, (19.29.26) reduces further to …

19: 13.22 Zeros

… ►From (13.14.2) and (13.14.3)

M_{\kappa,\mu}\left(z\right)

has the same zeros as

M\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)

and

W_{\kappa,\mu}\left(z\right)

has the same zeros as

U\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)

, hence the results given in §13.9 can be adopted. … ►For example, if

\mu(\geq 0)

is fixed and

\kappa(>0)

is large, then the

r

th positive zero

\phi_{r}

M_{\kappa,\mu}\left(z\right)

is given by ►

13.22.1

\phi_{r}=\frac{j_{2\mu,r}^{2}}{4\kappa}+j_{2\mu,r}O\left(\kappa^{-\frac{3}{2}}% \right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi_{r}$ : positive zeros and $j_{b,r}$ : positive zero of Bessel
Referenced by:: §13.22, §13.22
Permalink:: http://dlmf.nist.gov/13.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.22 and Ch.13

►where

j_{2\mu,r}

is the

r

th positive zero of the Bessel function

J_{2\mu}\left(x\right)

(§10.21(i)). …

20: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►If any lower argument in a

\mathit{6j}

symbol is

0

\tfrac{1}{2}

, or

1

, then the

\mathit{6j}

symbol has a simple algebraic form. … ►

34.5.9

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{1}&\frac{1}{2}(j_{2}+l_{2}+j_% {3}-l_{3})&\frac{1}{2}(j_{2}-l_{2}+j_{3}+l_{3})\\ l_{1}&\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{2}+l_{2}+j_{3}+l_{3% })\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(ii), §34.5 and Ch.34

… ►

34.5.11

{(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half., $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

… ►

34.5.16

(-1)^{j_{1}+j_{2}+j_{3}+j_{1}^{\prime}+j_{2}^{\prime}+l_{1}+l_{2}}\begin{% Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}^{\prime}&j_{2}^{\prime}&j_{% 3}\\ l_{1}&l_{2}&l_{3}^{\prime}\end{Bmatrix}=\sum_{j}(-1)^{l_{3}+l_{3}^{\prime}+j}(% 2j+1)\begin{Bmatrix}j_{1}&j_{1}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&j_{3}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{1}^{\prime}&j_{1}&l_{2}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&l_{1}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(vi)
Permalink:: http://dlmf.nist.gov/34.5.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

… ►

34.5.23

\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}=\sum_{m^{\prime}_{1}m^{\prime}_{2}m^{\prime}_{3% }}(-1)^{l_{1}+l_{2}+l_{3}+m^{\prime}_{1}+m^{\prime}_{2}+m^{\prime}_{3}}\begin{% pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(vi)
Permalink:: http://dlmf.nist.gov/34.5.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

…

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11—20 of 854 matching pages

11: 18.41 Tables

12: 1.3 Determinants, Linear Operators, and Spectral Expansions

13: 14.26 Uniform Asymptotic Expansions

14: 3.9 Acceleration of Convergence

15: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

16: 5.10 Continued Fractions

17: 11.14 Tables

18: 19.29 Reduction of General Elliptic Integrals

19: 13.22 Zeros

20: 34.5 Basic Properties: $\mathit{6j}$ Symbol

.竞菜足球_『wn4.com_』世界杯怎么下赌注2021_w6n2c9o_2022年12月1日4时27分49秒_3bnnfzbbz

11—20 of 854 matching pages

11: 18.41 Tables

12: 1.3 Determinants, Linear Operators, and Spectral Expansions

13: 14.26 Uniform Asymptotic Expansions

14: 3.9 Acceleration of Convergence

15: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

16: 5.10 Continued Fractions

17: 11.14 Tables

18: 19.29 Reduction of General Elliptic Integrals

19: 13.22 Zeros

20: 34.5 Basic Properties: 6 ⁢ j Symbol

20: 34.5 Basic Properties: $\mathit{6j}$ Symbol