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21: 10.41 Asymptotic Expansions for Large Order

… ►

U_{k+1}(p)=\tfrac{1}{2}p^{2}(1-p^{2})U_{k}^{\prime}(p)+\frac{1}{8}\int_{0}^{p}% (1-5t^{2})U_{k}(t)\,\mathrm{d}t,

… ►

U_{3}(p)=\tfrac{1}{4\;14720}\*(30375p^{3}-3\;69603p^{5}+7\;65765p^{7}-4\;25425% p^{9}),

►

V_{1}(p)=\tfrac{1}{24}(-9p+7p^{3}),

… ►The curve

E_{1}BE_{2}

in the

z

-plane is the upper boundary of the domain

\mathbf{K}

depicted in Figure 10.20.3 and rotated through an angle

-\tfrac{1}{2}\pi

. … ►This is because

A_{k}(\zeta)

and

\zeta^{-\frac{1}{2}}B_{k}(\zeta),k=0,1,\dotsc

, do not form an asymptotic scale (§2.1(v)) as

\zeta\to+\infty

; see Olver (1997b, pp. 422–425). …

22: 19.36 Methods of Computation

… ►If (19.36.1) is used instead of its first five terms, then the factor

(3r)^{-1/6}

in Carlson (1995, (2.2)) is changed to

(3r)^{-1/8}

. ►For both

R_{D}

and

R_{J}

the factor

(r/4)^{-1/6}

in Carlson (1995, (2.18)) is changed to

(r/5)^{-1/8}

when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: … ►Accurate values of

F\left(\phi,k\right)-E\left(\phi,k\right)

for

k^{2}

near 0 can be obtained from

R_{D}

by (19.2.6) and (19.25.13). … ►

E\left(\phi,k\right)

can be evaluated by using (19.25.7), and

R_{D}

by using (19.21.10), but cancellations may become significant. … ►Near these points there will be loss of significant figures in the computation of

R_{J}

R_{D}

. …

23: 3.2 Linear Algebra

… ►Assume that

\mathbf{A}

can be factored as in (3.2.5), but without partial pivoting. … ►where

\rho(\mathbf{A}\mathbf{A}^{\rm T})

is the largest of the absolute values of the eigenvalues of the matrix

\mathbf{A}\mathbf{A}^{\rm T}

; see §3.2(iv). … ►is called the characteristic polynomial of

\mathbf{A}

and its zeros are the eigenvalues of

\mathbf{A}

. … ►has the same eigenvalues as

\mathbf{A}

. … ►Many methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).

24: 16.25 Methods of Computation

… ►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).

25: 20 Theta Functions

…

26: 3.7 Ordinary Differential Equations

… ►where

f

g

, and

h

are analytic functions in a domain

D\subset\mathbb{C}

. … ►Assume that we wish to integrate (3.7.1) along a finite path

\mathscr{P}

from

z=a

z=b

in a domain

D

. The path is partitioned at

P+1

points labeled successively

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

, with

z_{0}=a

z_{P}=b

. … ►where

\mathbf{A}(\tau,z)

is the matrix … ►Let

\mathbf{A}_{P}

be the

(2P)\times(2P+2)

band matrix …

27: 19.27 Asymptotic Approximations and Expansions

… ►

§19.27(ii) $R_{F}\left(x,y,z\right)$

… ►

19.27.2

R_{F}\left(x,y,z\right)=\frac{1}{2\sqrt{z}}\left(\ln\frac{8z}{a+g}\right)\left% (1+O\left(\frac{a}{z}\right)\right),

a/z\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ and $g$
Referenced by:: §19.20(iii), §19.27(ii)
Permalink:: http://dlmf.nist.gov/19.27.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.27(ii), §19.27 and Ch.19

… ►

19.27.6

R_{G}\left(0,y,z\right)={\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln\left(% \frac{16z}{y}\right)-1\right)\*\left(1+O\left(\frac{y}{z}\right)\right)},

y/z\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $\ln\NVar{z}$ : principal branch of logarithm function
Permalink:: http://dlmf.nist.gov/19.27.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.27(iii), §19.27 and Ch.19

►

§19.27(iv) $R_{D}\left(x,y,z\right)$

… ►

19.27.7

R_{D}\left(x,y,z\right)=\frac{3}{2z^{3/2}}\left(\ln\left(\frac{8z}{a+g}\right)% -2\right)\left(1+O\left(\frac{a}{z}\right)\right),

a/z\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ and $g$
Referenced by:: §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.27.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.27(iv), §19.27 and Ch.19

…

28: 19.29 Reduction of General Elliptic Integrals

… ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the

8+8+12=28

formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking

x^{2}

as the variable of integration in 3. …where the arguments of the

R_{D}

function are, in order,

(a-b)(u-c)

(b-c)(a-u)

(a-b)(b-c)

. … ►The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting

x^{2}=t

in some cases). … ►where …(The variables of

R_{F}

are real and nonnegative unless both

Q

’s have real zeros and those of

Q_{1}

interlace those of

Q_{2}

.) …

29: 3.5 Quadrature

… ►If

f\in C^{2m+2}[a,b]

, then the remainder

E_{n}(f)

in (3.5.2) can be expanded in the form … ►About

2^{9}=512

function evaluations are needed. … ►with weight function

w(x)

, is one for which

E_{n}(f)=0

whenever

f

is a polynomial of degree

\leq n-1

. … ►For further information, see Mason and Handscomb (2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74–92), and Clenshaw and Curtis (1960). … ►Table 3.5.21 supplies cubature rules, including weights

w_{j}

, for the disk

D

, given by

x^{2}+y^{2}\leq h^{2}

: …

30: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

► ►►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8	9	10
…
8	1	8	28	56	70	56	28	8	1
…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8
…
1	1	2	3	4	5	6	7	8	9
…
8	1	9	45	165	495	1287	3003	6435	12870

► …