remainder terms

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11: 10.17 Asymptotic Expansions for Large Argument

… ►Then the remainder associated with the sum ${\sum }_{k=0}^{\mathrm{\ell }-1}{(-1)}^k{a}_{2k}(\nu )z^{-2k}$ does not exceed the first neglected term in absolute value and has the same sign provided that $\mathrm{\ell }\ge \max (\frac{1}{2}\nu -\frac{1}{4},1)$. … ►If these expansions are terminated when $k=\mathrm{\ell }-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\mathrm{\ell }\ge \max (\nu -\frac{1}{2},1)$. … ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

12: Bibliography K

… ►

An improvement of the remainder term in the divisor problem. Mat. Zametki 6, pp. 545–554 (Russian).

ⓘ

Notes:

In Russian. English translation: Math. Notes 6(1969), no. 5, pp. 784–791.

External Links:

Document, MathReview (A. I. Vinogradov), ZentralBlatt

Referenced by:

§27.11, Erratum (V1.1.8) for Section 27.11.

Encodings:

BibTeX

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13: 9.7 Asymptotic Expansions

… ►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). …

14: 2.6 Distributional Methods

… ►An important asset of the distribution method is that it gives explicit expressions for the remainder terms associated with the resulting asymptotic expansions. … ►Furthermore, since $f_{n,n}^{(n)}(t)=f_n(t)$, it follows from (2.6.37) that the remainder terms $t^{\mu -1}\ast f_n$ in the last two equations can be associated with a locally integrable function in $(0,\mathrm{\infty })$. …

15: 2.10 Sums and Sequences

… ►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\ge \frac{1}{2}(\alpha +1)$. …

16: 2.7 Differential Equations

… ►where ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ are constants, and the $J$th remainder terms in the sums are $O\left(\mathrm{\Gamma }\left(s+{\mu }_2-{\mu }_1-J\right)\right)$ and $O\left(\mathrm{\Gamma }\left(s+{\mu }_1-{\mu }_2-J\right)\right)$, respectively (Olver (1994a)). …

17: 3.5 Quadrature

… ►The remainder is given by …

18: 9.8 Modulus and Phase

… ►The remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n\ge 0$ for (9.8.20), (9.8.22) and (9.8.23), and $n\ge 1$ for (9.8.21). …

19: 10.18 Modulus and Phase Functions

… ►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

20: 22.9 Cyclic Identities

… ►In the remainder of this section the rank of an identity is the largest number of elliptic function factors in any term of the identity. …