Romberg integration

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31—40 of 150 matching pages

31: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

§1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$

… ►For a Lebesgue–Stieltjes measure

\,\mathrm{d}\alpha

X

let

L^{2}\left(X,\,\mathrm{d}\alpha\right)

be the space of all Lebesgue–Stieltjes measurable complex-valued functions on

X

which are square integrable with respect to

\,\mathrm{d}\alpha

, … ►We integrate by parts twice giving: … ►Eigenfunctions corresponding to the continuous spectrum are non-

L^{2}

functions. … ►The well must be deep and broad enough to allow existence of such

L^{2}

discrete states. …

32: 10.74 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. ►In the interval

0<x<\nu

J_{\nu}\left(x\right)

needs to be integrated in the forward direction and

Y_{\nu}\left(x\right)

in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). In the interval

\nu<x<\infty

either direction of integration can be used for both functions. ►Similarly, to maintain stability in the interval

0<x<\infty

the integration direction has to be forwards in the case of

I_{\nu}\left(x\right)

and backwards in the case of

K_{\nu}\left(x\right)

, with initial values obtained in an analogous manner to those for

J_{\nu}\left(x\right)

and

Y_{\nu}\left(x\right)

. …

33: 13.29 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. ►For

M\left(a,b,z\right)

and

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

this means that in the sector

|\operatorname{ph}z|\leq\pi

we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). … ►In the sector

|\operatorname{ph}z|<\tfrac{1}{2}\pi

the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). On the rays

\operatorname{ph}z=\pm\tfrac{1}{2}\pi

, integration can proceed in either direction. …

34: 15.19 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions. However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. …

35: 36.14 Other Physical Applications

… ►Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. …

36: 1.8 Fourier Series

… ►where

f(x)

is square-integrable on

[-\pi,\pi]

and

a_{n},b_{n},c_{n}

are given by (1.8.2), (1.8.4). If

g(x)

is also square-integrable with Fourier coefficients

a_{n}^{\prime},b_{n}^{\prime}

c_{n}^{\prime}

then … ►Let

f(x)

be an absolutely integrable function of period

2\pi

, and continuous except at a finite number of points in any bounded interval. … ►

§1.8(iii) Integration and Differentiation

… ►Suppose that

f(x)

is twice continuously differentiable and

f(x)

and

\left|f^{\prime\prime}(x)\right|

are integrable over

(-\infty,\infty)

. …

37: 28.34 Methods of Computation

… ►

(a)

Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

… ►

(b)

Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

…

38: Philip J. Davis

… ►An interesting anecdote told by Davis reveals that he and mathematician Philip Rabinowitz were dubbed “Heroes of the SEAC” when their Gaussian integration code executed correctly on its first run. … ►Davis also co-authored a second Chapter, “Numerical Interpolation, Differentiation, and Integration” with Ivan Polonsky. …

39: 31.9 Orthogonality

… ►The integration path begins at

z=\zeta

, encircles

z=1

once in the positive sense, followed by

z=0

once in the positive sense, and so on, returning finally to

z=\zeta

. The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). … ►

31.9.5

\int_{\mathcal{L}_{1}}\int_{\mathcal{L}_{2}}\rho(s,t)w_{1}(s)w_{1}(t)w_{2}(s)w% _{2}(t)\,\mathrm{d}s\,\mathrm{d}t=0,

|n_{1}-n_{2}|+|m_{1}-m_{2}|\neq 0

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer, $\rho(s,t)$ : weight function and $\mathcal{L}_{1}$ , $\mathcal{L}_{2}$ : integration paths
Permalink:: http://dlmf.nist.gov/31.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(ii), §31.9 and Ch.31

… ►and the integration paths

\mathcal{L}_{1}

\mathcal{L}_{2}

are Pochhammer double-loop contours encircling distinct pairs of singularities

\{0,1\}

\{0,a\}

\{1,a\}

. …

40: 8.6 Integral Representations

… ►where the integration path passes above or below the pole at

t=1

, according as upper or lower signs are taken. … ►In (8.6.10)–(8.6.12),

c

is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at

s=a

, in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at

s=0,1,2,\ldots

. …