on finite point sets

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21: 3.5 Quadrature

… ►Let

\{p_{n}\}

denote the set of monic polynomials

p_{n}

of degree

n

(coefficient of

x^{n}

equal to

1

) that are orthogonal with respect to a positive weight function

w

on a finite or infinite interval

(a,b)

; compare §18.2(i). … ►To avoid cancellation we try to deform the path to pass through a saddle point in such a way that the maximum contribution of the integrand is derived from the neighborhood of the saddle point. … ►with saddle point at

t=1

, and when

c=1

the vertical path intersects the real axis at the saddle point. … ►If

f

is meromorphic, with poles near the saddle point, then the foregoing method can be modified. … ►The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. …

22: 18.1 Notation

… ► ►►►

$x, y, t$	real variables.
…
$w_{x}$	weights $(>0)$ at points $x\in X$ of a finite or countably infinite subset of $\mathbb{R}$ .
…

… ►

\delta_{x}\left(f(x)\right)=\left(f(x+\tfrac{1}{2}\mathrm{i})-f(x-\tfrac{1}{2}% \mathrm{i})\right)/\mathrm{i},

…

23: 32.11 Asymptotic Approximations for Real Variables

… ►

32.11.2

\phi(x)=(24)^{1/4}\left(\tfrac{4}{5}|x|^{5/4}-\tfrac{5}{8}d^{2}\ln|x|\right),

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real, $\phi(z)$ : function and $d$ : constant
Referenced by:: §32.11(i)
Permalink:: http://dlmf.nist.gov/32.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §32.11(i), §32.11 and Ch.32

… ►

32.11.6

w_{k}(x)=d|x|^{-1/4}\sin\left(\phi(x)-\theta_{0}\right)+o\left(|x|^{-1/4}% \right),

ⓘ

Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $x$ : real, $k$ : real, $\phi(z)$ : function, $d$ : constant and $\theta_{0}$ : constant
Permalink:: http://dlmf.nist.gov/32.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §32.11(ii), §32.11 and Ch.32

… ►If

|k|>1

, then

w_{k}(x)

has a pole at a finite point

x=c_{0}

, dependent on

k

, and … ►where

\lambda

is an arbitrary constant such that

-1/\pi<\lambda<1/\pi

, and … ►In terms of the parameter

k

that is used in these figures

h=2^{3/2}k^{2}

. …

24: 18.3 Definitions

… ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). … ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials

T_{n}\left(x\right)

n=0,1,\dots,N

, are orthogonal on the discrete point set comprising the zeros

x_{N+1,n},n=1,2,\dots,N+1

, of

T_{N+1}\left(x\right)

: … ►For

-1-\beta>\alpha>-1

a finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

is orthogonal on

(1,\infty)

with weight function

w(x)=(x-1)^{\alpha}(x+1)^{\beta}

. For

\nu\in\mathbb{R}

and

N>-\tfrac{1}{2}

a finite system of Jacobi polynomials

P^{(-N-1+\mathrm{i}\nu,-N-1-\mathrm{i}\nu)}_{n}\left(\mathrm{i}x\right)

(called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on

(-\infty,\infty)

with

w(x)=\left(1+x^{2}\right)^{-N-1}{\mathrm{e}}^{2\nu\operatorname{arctan}x}

. … ►However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. …

25: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii

\rho

and

r

, respectively, and may be used to compute the regular and irregular solutions. … ►Inside the turning points, that is, when

\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)

, there can be a loss of precision by a factor of approximately

|G_{\ell}|^{2}

. … ►WKBJ approximations (§2.7(iii)) for

\rho>\rho_{\operatorname{tp}}\left(\eta,\ell\right)

are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. …A set of consistent second-order WKBJ formulas is given by Burgess (1963: in Eq. … ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for

F_{0}

and

G_{0}

in the region inside the turning point:

\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)

26: 3.7 Ordinary Differential Equations

… ►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

. … ►Then to compute

w(z)

in a stable manner we solve the set of equations (3.7.5) simultaneously for

j=0,1,\dots,P

, as follows. … ►Let

(a,b)

be a finite or infinite interval and

q(x)

be a real-valued continuous (or piecewise continuous) function on the closure of

(a,b)

. … ►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …

27: 2.10 Sums and Sequences

… ►for any real constant

\alpha

and the set of all positive integers

j

, we derive … ►However, if

r

is finite and

f(z)

has algebraic or logarithmic singularities on

|z|=r

, then Darboux’s method is usually easier to apply. … ►The singularities of

f(z)

on the unit circle are branch points at

z=e^{\pm i\alpha}

. To match the limiting behavior of

f(z)

at these points we set …and in the supplementary conditions we may set

m=1

. …

28: 2.4 Contour Integrals

… ►

§2.4(iv) Saddle Points

… ►

§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

… ►

§2.4(vi) Other Coalescing Critical Points

… ►For a coalescing saddle point, a pole, and a branch point see Ciarkowski (1989). For many coalescing saddle points see §36.12. …

29: 1.13 Differential Equations

… ►A solution becomes unique, for example, when

w

and

\ifrac{\mathrm{d}w}{\mathrm{d}z}

are prescribed at a point in

D

. … ►Then at each

z\in D

w

\ifrac{\partial w}{\partial z}

and

\ifrac{{\partial}^{2}w}{{\partial z}^{2}}

are analytic functions of

u

. … ►

Transformation of the Point at Infinity

… ►on a finite interval

[a,b]\subset\mathbb{R}

, this is then a regular Sturm-Liouville system. … ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues,

\lambda

; (ii) the corresponding (real) eigenfunctions,

u(x)

and

w(t)

, have the same number of zeros, also called nodes, for

t\in(0,c)

as for

x\in(a,b)

; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

30: 1.14 Integral Transforms

… ►If also

\lim_{t\to 0+}f(t)/t

exists, then … ►where

A_{p}=\tan\left(\tfrac{1}{2}\pi/p\right)

when

1 <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math>, or <math class=

when

p\geq 2

. … ►If the integral converges, then it converges uniformly in any compact domain in the complex

s

-plane not containing any point of the interval

(-\infty,0]

. … ►If

f(t)

is absolutely integrable on

[0,R]

for every finite

R

, and the integral (1.14.47) converges, then … ►

Table 1.14.3: Fourier sine transforms.

► ►►►

$f(t)$	$\sqrt{\dfrac{2}{\pi}}\displaystyle\int^{\infty}_{0}\!\!\!f(t)\sin\left(xt% \right)\,\mathrm{d}t$ ,	$x>0$
…
$t^{-1/2}$	$x^{-1/2}$
…

► …