DLMF: Untitled Document

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: … ►where

\xi_{0}

and

\xi_{1}\in I

. ►For the values of

n_{0}

and

n_{1}

used in the formulas below … ►For partial derivatives we use the notation

u_{t,s}=u(x_{0}+th,y_{0}+sh)

. …

… ►

•

Pearson (1965) tabulates the function $I(u,p)$ ( $=P\left(p+1,u\right)$ ) for $p=-1(.05)0(.1)5(.2)50$ , $u=0(.1)u_{p}$ to 7D, where $I(u,u_{p})$ rounds off to 1 to 7D; also $I(u,p)$ for $p=-0.75(.01)-1$ , $u=0(.1)6$ to 5D.

… ►

•

Pearson (1968) tabulates $I_{x}\left(a,b\right)$ for $x=0.01(.01)1$ , $a,b=0.5(.5)11(1)50$ , with $b\leq a$ , to 7D.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

►

•

Stankiewicz (1968) tabulates $E_{n}\left(x\right)$ for $n=1(1)10$ , $x=0.01(.01)5$ to 7D.

►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

… ► ►►►►►►►

$g, h$	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).)
…

…

… ►The choice of

x_{0}

here is critical. … ►Let

x_{0}

and

x_{1}

be such that

f_{0}=f(x_{0})

and

f_{1}=f(x_{1})

have opposite signs. …We continue with

x_{2}

and either

x_{0}

or

x_{1}

, depending which of

f_{0}

and

f_{1}

is of opposite sign to

f(x_{2})

, and so on. … ►Whether or not

f_{0}

and

f_{1}

have opposite signs,

x_{2}

is computed as in (3.8.6). … ►We construct sequences

q_{j}

and

r_{j}

,

j=n+1,n,\dots,0

, from

q_{n+1}=r_{n+1}=0

,

q_{n}=r_{n}=a_{n}

, and for

j\leq n-1

, …

… ►Leading terms of the power series for

a_{m}\left(q\right)

and

b_{m}\left(q\right)

for

m\leq 6

are: … ►The coefficients of the power series of

a_{2n}\left(q\right)

,

b_{2n}\left(q\right)

and also

a_{2n+1}\left(q\right)

,

b_{2n+1}\left(q\right)

are the same until the terms in

q^{2n-2}

and

q^{2n}

, respectively. … ►Numerical values of the radii of convergence

\rho_{n}^{(j)}

of the power series (28.6.1)–(28.6.14) for

n=0,1,\dots,9

are given in Table 28.6.1. Here

j=1

for

a_{2n}\left(q\right)

,

j=2

for

b_{2n+2}\left(q\right)

, and

j=3

for

a_{2n+1}\left(q\right)

and

b_{2n+1}\left(q\right)

. … ►

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

…

… ►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).

… ►Using (3.5.10) with

h=30/4=7.5

we obtain

G_{7}(h)

with 14 correct digits. … ►The nodes

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

are prescribed, and the weights

w_{k}

and error term

E_{n}(f)

are found by integrating the product of the Lagrange interpolation polynomial of degree

n-1

and

w(x)

. … ►The nodes

x_{k}

and weights

w_{k}

are known explicitly: … ►Let

\mathbf{v}_{k}

denote the normalized eigenvector of

\mathbf{J}_{n}

corresponding to the eigenvalue

x_{k}

. … ►The monic version

p_{n}(x)

and orthonormal version

q_{n}(x)

of a classical orthogonal polynomial are obtained by dividing the orthogonal polynomial by

k_{n}

respectively

\sqrt{h_{n}}

, with

k_{n}

and

h_{n}

as in Table 18.3.1. …

… ►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 $p$

…

… ►

•

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.

… ►

•

Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

… ►

•

Harvard University (1945) tabulates the real and imaginary parts of $h_{1}(z)$ , $h_{1}^{\prime}(z)$ , $h_{2}(z)$ , $h_{2}^{\prime}(z)$ for $-x_{0}\leq\Re z\leq x_{0}$ , $0\leq\Im z\leq y_{0}$ , $|x_{0}+iy_{0}|<6.1$ , with interval 0.1 in $\Re z$ and $\Im z$ . Precision is 8D. Here $h_{1}(z)=-2^{4/3}3^{1/6}i\operatorname{Ai}\left(e^{-\pi i/3}z\right)$ , $h_{2}(z)=2^{4/3}3^{1/6}i\operatorname{Ai}\left(e^{\pi i/3}z\right)$ .

… ►

•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

… ►

•

National Bureau of Standards (1958) tabulates $A_{0}(x)\equiv\pi\operatorname{Hi}\left(-x\right)$ and $-A_{0}^{\prime}(x)\equiv\pi\operatorname{Hi}'\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$ ; $\int_{0}^{x}A_{0}(t)\,\mathrm{d}t$ for $x=0.5,1(1)11$ . Precision is 8D.

…

.94世界杯马拉多纳_『wn4.com_』2014年世界杯德国vs阿根廷_w6n2c9o_2022年11月30日9时23分57秒_yo6cq20ae

21—30 of 785 matching pages

21: 3.4 Differentiation

22: 8.26 Tables

23: 21.1 Special Notation

24: 3.8 Nonlinear Equations

25: 28.6 Expansions for Small $q$

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

26: 24.20 Tables

27: 3.5 Quadrature

28: 24.19 Methods of Computation

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

29: 11.14 Tables

30: 9.18 Tables

.94世界杯马拉多纳_『wn4.com_』2014年世界杯德国vs阿根廷_w6n2c9o_2022年11月30日9时23分57秒_yo6cq20ae

21—30 of 785 matching pages

21: 3.4 Differentiation

22: 8.26 Tables

23: 21.1 Special Notation

24: 3.8 Nonlinear Equations

25: 28.6 Expansions for Small q

§28.6(ii) Functions ce n and se n

26: 24.20 Tables

27: 3.5 Quadrature

28: 24.19 Methods of Computation

§24.19(ii) Values of B n Modulo p

29: 11.14 Tables

30: 9.18 Tables

25: 28.6 Expansions for Small $q$

§28.6(ii) Functions $\operatorname{ce}_{n}$ and $\operatorname{se}_{n}$

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