.1966年世界杯决赛悬案_『wn4.com_』国际象棋世界杯第十轮_w6n2c9o_2022年11月28日19时19分39秒_swou6u2qo_cc

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21—30 of 782 matching pages

22: 28.29 Definitions and Basic Properties

… ►The basic solutions

w_{\mbox{\tiny I}}(z,\lambda)

w_{\mbox{\tiny II}}(z,\lambda)

are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). … ►where the function

P_{\nu}(z)

\pi

-periodic. … ►If

\nu

(\neq 0,1)

is a solution of (28.29.9), then

F_{\nu}(z)

F_{-\nu}(z)

comprise a fundamental pair of solutions of Hill’s equation. … ►In the symmetric case

Q(z)=Q(-z)

w_{\mbox{\tiny I}}(z,\lambda)

is an even solution and

w_{\mbox{\tiny II}}(z,\lambda)

is an odd solution; compare §28.2(ii). …The

\pi

-periodic or

\pi

-antiperiodic solutions are multiples of

w_{\mbox{\tiny I}}(z,\lambda),w_{\mbox{\tiny II}}(z,\lambda)

, respectively. …

$g, h$	positive integers.
…
$a_{j}$	$j$ th element of vector $\mathbf{a}$ .
…
$\operatorname{diag}\mathbf{A}$	Transpose of $[A_{11},A_{22},\dots,A_{gg}]$ .
…
$\mathbf{J}_{2g}$	$\begin{bmatrix}\boldsymbol{{0}}_{g}&\mathbf{I}_{g}\\ -\mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$ .
…
$S_{1}S_{2}$	set of all elements of the form “ $\mbox{element of $S_{1}$}\times\mbox{element of $S_{2}$}$ ”.
$S_{1}/S_{2}$	set of all elements of $S_{1}$ , modulo elements of $S_{2}$ . Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$ . (For an example see §20.12(ii).)
…

24: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Zhang and Jin (1996, p. 199) tabulates the real and imaginary parts of the first 15 conjugate pairs of complex zeros of $Y_{0}\left(z\right)$ , $Y_{1}\left(z\right)$ , $Y_{1}'\left(z\right)$ and the corresponding values of $Y_{1}\left(z\right)$ , $Y_{0}\left(z\right)$ , $Y_{1}\left(z\right)$ , respectively, 10D.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 10D; $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 8D.

… ►

•

Achenbach (1986) tabulates $I_{0}\left(x\right)$ , $I_{1}\left(x\right)$ , $K_{0}\left(x\right)$ , $K_{1}\left(x\right)$ , $x=0(.1)8$ , 19D or 19–21S.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 7D; $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 6D.

…

25: 3.3 Interpolation

… ►The nodes or abscissas

z_{k}

are real or complex; function values are

f_{k}=f(z_{k})

. … ►Let

c_{n}

be defined by … ►The divided differences of

f

relative to a sequence of distinct points

z_{0},z_{1},z_{2},\dots

are defined by …If

f

and the

z_{k}

(

=x_{k}

) are real, and

f

n

times continuously differentiable on a closed interval containing the

x_{k}

, then … ►By using this approximation to

x

as a new point,

x_{3}=x

, and evaluating

\left[f_{0},f_{1},f_{2},f_{3}\right]x=1.12388\;6190

, we find that

x=-2.33810\;7409

, with 9 correct digits. …

26: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

… ►

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

►

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

… ►where

m_{1},m_{2},\dots,m_{r}

are arbitrary nonnegative integers. …

27: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►To emphasize this property this set of functions is denoted by … ►The eigenvalues

q_{m}

satisfy the continued-fraction equation …in which

P_{j},Q_{j},R_{j}

are as in §31.3(i). … ►with

(s_{1},s_{2})\in\{0,1,a,\infty\}

, denotes a set of solutions of (31.2.1), each of which is analytic at

s_{1}

and

s_{2}

. The set

q_{m}

depends on the choice of

s_{1}

and

s_{2}

. …

28: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of

\sum_{k=1}^{m}k^{n}

m=1(1)100

n=1(1)10

;

\sum_{k=1}^{\infty}k^{-n}

\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}

\sum_{k=0}^{\infty}(2k+1)^{-n}

n=1,2,\dotsc

, 20D;

\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}

n=1,2,\dotsc

, 18D. ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

29: 24.19 Methods of Computation

… ►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

to be computed by recurrence. …For example, the tangent numbers

T_{n}

can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. … ►If

\widetilde{N}_{2n}

denotes the right-hand side of (24.19.1) but with the second product taken only for

p\leq\left\lfloor(\pi e)^{-1}2n\right\rfloor+1

, then

N_{2n}=\left\lceil\widetilde{N}_{2n}\right\rceil

for

n\geq 2

. … ►For algorithms for computing

B_{n}

E_{n}

B_{n}\left(x\right)

, and

E_{n}\left(x\right)

see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

…

30: 11.14 Tables

… ►

•

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.

… ►

•

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.

►

•

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

… ►

•

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.

.1966年世界杯决赛悬案_『wn4.com_』国际象棋世界杯第十轮_w6n2c9o_2022年11月28日19时19分39秒_swou6u2qo_cc

21—30 of 782 matching pages

21: 5.10 Continued Fractions

22: 28.29 Definitions and Basic Properties

23: 21.1 Special Notation

24: 10.75 Tables

25: 3.3 Interpolation

26: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

27: 31.4 Solutions Analytic at Two Singularities: Heun Functions

28: 24.20 Tables

29: 24.19 Methods of Computation

§24.19(ii) Values of $B_{n}$ Modulo $p$

30: 11.14 Tables

.1966年世界杯决赛悬案_『wn4.com_』国际象棋世界杯第十轮_w6n2c9o_2022年11月28日19时19分39秒_swou6u2qo_cc

21—30 of 782 matching pages

21: 5.10 Continued Fractions

22: 28.29 Definitions and Basic Properties

23: 21.1 Special Notation

24: 10.75 Tables

25: 3.3 Interpolation

26: 17.7 Special Cases of Higher ϕ s r Functions

§17.7(i) ϕ 2 2 Functions

q -Analog of Bailey’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Gauss’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Dixon’s F 2 3 ⁡ ( 1 ) Sum

27: 31.4 Solutions Analytic at Two Singularities: Heun Functions

28: 24.20 Tables

29: 24.19 Methods of Computation

§24.19(ii) Values of B n Modulo p

30: 11.14 Tables

26: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

§24.19(ii) Values of $B_{n}$ Modulo $p$