re-expansion%20of%20remainder%20terms

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1: 12.9 Asymptotic Expansions for Large Variable

… ►

§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

►Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). …

2: 6.12 Asymptotic Expansions

… ►For these and other error bounds see Olver (1997b, pp. 109–112) with

\alpha=0

. ►For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv), with

p=1

. …If the expansion is terminated at the

n

th term, then the remainder term is bounded by

1+\chi(n+1)

times the next term. … ►The remainder terms are given by …When

|\operatorname{ph}z|\leq\tfrac{1}{4}\pi

, these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when

\operatorname{ph}z=0

. …

3: 2.11 Remainder Terms; Stokes Phenomenon

… ►

§2.11(iii) Exponentially-Improved Expansions

… ►If we permit the use of nonelementary functions as approximants, then even more powerful re-expansions become available. … ►For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … ►Often the process of re-expansion can be repeated any number of times. … ►For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.

4: 8.22 Mathematical Applications

… ►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. …

5: 7.12 Asymptotic Expansions

… ►When

|\operatorname{ph}z|\leq\frac{1}{4}\pi

the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when

\operatorname{ph}z=0

. When

\frac{1}{4}\pi\leq|\operatorname{ph}z|<\frac{1}{2}\pi

the remainder terms are bounded in magnitude by

\csc\left(2|\operatorname{ph}z|\right)

times the first neglected terms. … ►For re-expansions of the remainder terms leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv) and use (7.11.3). (Note that some of these re-expansions themselves involve the complementary error function.) … ►The remainder terms are given by …

6: 11.6 Asymptotic Expansions

… ►

§11.6(i) Large $|z|$ , Fixed $\nu$

… ►If the series on the right-hand side of (11.6.1) is truncated after

m(\geq 0)

terms, then the remainder term

R_{m}(z)

O\left(z^{\nu-2m-1}\right)

. If

\nu

is real,

z

is positive, and

m+\tfrac{1}{2}-\nu\geq 0

, then

R_{m}(z)

is of the same sign and numerically less than the first neglected term. … ►For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … ►

c_{3}(\lambda)=20\lambda^{-6}-4\lambda^{-4},

…

7: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of

g_{k}

up to

g_{20}

. … ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at

k=n-1

(

k\geq 0

) and

z

is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If

z

is complex, then the remainder terms are bounded in magnitude by

{\sec}^{2n}\left(\tfrac{1}{2}\operatorname{ph}z\right)

for (5.11.1), and

{\sec}^{2n+1}\left(\tfrac{1}{2}\operatorname{ph}z\right)

for (5.11.2), times the first neglected terms. … ►For the remainder term in (5.11.3) write … ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). …

8: 9.7 Asymptotic Expansions

… ►Numerical values of

\chi(n)

are given in Table 9.7.1 for

n=1(1)20

to 2D. … ►In (9.7.9)–(9.7.12) the

n

th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign. … ►The

n

th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by … ►For re-expansions of the remainder terms in (9.7.7)–(9.7.14) combine the results of this section with those of §9.2(v) and their differentiated forms, as in §9.7(iv). ►For higher re-expansions of the remainder terms see Olde Daalhuis (1995, 1996), and Olde Daalhuis and Olver (1995a). …

9: 28.8 Asymptotic Expansions for Large $q$

… ►

28.8.1

\rselection{a_{m}\left(h^{2}\right)\\ b_{m+1}\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h% }(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{17}h^{3}}(33s^% {5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-% \frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $m$ : integer and $h$ : parameter
A&S Ref:: 20.2.30 (in different form)
Referenced by:: §28.8(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(i), §28.8 and Ch.28

… ►The approximations are expressed in terms of Whittaker functions

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

and

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

with

\mu=\tfrac{1}{4}

; compare §2.8(vi). …With additional restrictions on

z

, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions

\operatorname{me}_{\nu}\left(z,q\right)

(§28.12(ii)) and modified Mathieu functions

{\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)

(§28.20(iii)). …

10: 10.40 Asymptotic Expansions for Large Argument

… ►The general terms in (10.40.6) and (10.40.7) can be written down by analogy with (10.18.17), (10.18.19), and (10.18.20). … ►Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that

\ell\geq\max(|\nu|-\tfrac{1}{2},1)

. ►For the error term in (10.40.1) see §10.40(iii). … ►

10.40.11

|R_{\ell}(\nu,z)|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\infty}\left(t^{-\ell}% \right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,\infty}\left(t^{-1}% \right)\right),

ⓘ

Defines:: $R_{\ell}(\nu,z)$ : remainder (locally)
Symbols:: $\exp\NVar{z}$ : exponential function, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation, $z$ : complex variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
Permalink:: http://dlmf.nist.gov/10.40.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §10.40(iii), §10.40 and Ch.10

… ►For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).