remainder terms

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21—28 of 28 matching pages

21: Bibliography M

… ►

P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function

{\rm Ai}(x)

. J. Comput. Phys. 99 (2), pp. 337–340.

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Notes:: There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$ .
External Links:: ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

J. P. McClure and R. Wong (1978) Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

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External Links:: ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt
Referenced by:: §2.6(ii).
Encodings:: BibTeX

►

J. P. McClure and R. Wong (1979) Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.

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External Links:: ISSN 0020-2932, Document, MathReview, ZentralBlatt
Referenced by:: §2.6(iii).
Encodings:: BibTeX

… ►

H. J. W. Müller (1966b) Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.

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External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §29.11, §29.7(ii).
Encodings:: BibTeX

…

22: 8.11 Asymptotic Approximations and Expansions

… ►Then as

z\to\infty

with

a

and

n

fixed … ►If

a

is real and

z

(

=x

) is positive, then

R_{n}(a,x)

is bounded in absolute value by the first neglected term

u_{n}/x^{n}

and has the same sign provided that

n\geq a-1

. … ►This reference also contains explicit formulas for

b_{k}(\lambda)

in terms of Stirling numbers and for the case

\lambda>1

an asymptotic expansion for

b_{k}(\lambda)

k\to\infty

. … ►See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. … ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …

23: 8.27 Approximations

… ►

•

Luke (1975, §4.3) gives Padé approximation methods, combined with a detailed analysis of the error terms, valid for real and complex variables except on the negative real $z$ -axis. See also Temme (1994b, §3).

… ►

•

Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_{1}\left(z\right)$ and $z^{-1}\int_{0}^{z}t^{-1}(1-e^{-t})\,\mathrm{d}t$ for complex $z$ with $\left|\operatorname{ph}z\right|\leq\pi$ .

…

24: 3.11 Approximation Techniques

… ►where

x_{\ell}=\cos\left(\pi\ell/n\right)

and the double prime means that the first and last terms are to be halved. … ►Here the single prime on the summation symbol means that the first term is to be halved. … ►More precisely, it is known that for the interval

[a,b]

, the ratio of the maximum value of the remainder … ►Let

c_{n}T_{n}\left(x\right)

be the last term retained in the truncated series. … ►Multivariate functions can also be approximated in terms of multivariate polynomial splines. …

25: 18.28 Askey–Wilson Class

… ►In the remainder of this section the Askey–Wilson class OP’s are defined by their

q

-hypergeometric representations, followed by their orthogonal properties. … ►Define dual parameters

\tilde{a},\tilde{b},\tilde{c},\tilde{d}

in terms of

a, b, c, d

by … ►More generally, if

|ab|\leq 1

instead of

|a|,|b|\leq 1

, discrete terms need to be added to the right-hand side of (18.28.8); see Koekoek et al. (2010, Eq. (14.8.3)). …

26: 18.39 Applications in the Physical Sciences

… ►However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of

\mathcal{H}

form a discrete, normed, and complete basis for a Hilbert space. … ►

a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

… ►The solution, (18.39.29), of the spherical radial equation (18.39.28), now expressed in terms of the Bohr quantum number

n

, is … ►

d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

… ►These same solutions are expressed here in terms of Laguerre and Pollaczek OP’s. …

27: 1.5 Calculus of Two or More Variables

… ►

1.5.18

f(a+\lambda,b+\mu)=f+\left(\lambda\frac{\partial}{\partial x}+\mu\frac{% \partial}{\partial y}\right)f+\dots+\frac{1}{n!}\left(\lambda\frac{\partial}{% \partial x}+\mu\frac{\partial}{\partial y}\right)^{n}f+R_{n},

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Symbols:: $!$ : factorial (as in $n!$ ), $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $n$ : nonnegative integer and $R_{\NVar{n}}$ : remainder
Referenced by:: §1.5(iii)
Permalink:: http://dlmf.nist.gov/1.5.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §1.5(iii), §1.5 and Ch.1

… ►and the second order term in (1.5.18) is positive definite (negative definite), that is, …

28: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Focus is now placed on second order differential operators as these are the subject of the remainder of §1.18. … ►Ignoring the boundary value terms it follows that … ►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that

\mathcal{D}(T)=\mathcal{D}({T}^{*})

, as

f(x)

and

g(x)

satisfy the same boundary conditions and thus define the same domains. … ►Other choices of boundary conditions, identical for

f(x)

and

g(x)

, and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of

T

. … ►Eigenvalues and eigenfunctions of

T

, self-adjoint extensions of

\mathcal{L}

with well defined boundary conditions, and utilization of such eigenfunctions for expansion of wide classes of

L^{2}

functions, will be the focus of the remainder of this section. …