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11: 10.41 Asymptotic Expansions for Large Order

… ►Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with $z$ replaced by $\nu z$, up to and including the term in $z^{-(\mathrm{\ell }-1)}$. … ►Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. …This is done by re-expansion with the aid of (10.20.10), (10.20.11), and §10.41(ii), followed by comparison with (10.17.5) and (10.17.6), with $z$ replaced by $\nu z$. …

12: 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 $\mathrm{\ell }\ge \max (\nu -\frac{1}{2},1)$. … ►

10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

$R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.17(iv), Erratum (V1.0.10) for Equation (10.17.14)

Permalink:

http://dlmf.nist.gov/10.17.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

… ►

10.17.18 $$R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)=O\left({\mathrm{e}}^{-2|z|}{z}^{-m}\right),$$ $|\mathrm{ph}\left(z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}\right)|\le \pi$.

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Defines:

$R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally)

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.17.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(v), §10.17 and Ch.10

►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

13: 20 Theta Functions

Chapter 20 Theta Functions

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14: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. … ►Their product $m$ has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

15: 8 Incomplete Gamma and Related
Functions

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16: 28 Mathieu Functions and Hill’s Equation

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17: 8.26 Tables

… ►

Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

… ►

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

18: 23 Weierstrass Elliptic and Modular
Functions

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19: 6.16 Mathematical Applications

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6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

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Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

6.16.3 $$R_n(x)=\frac{1}{2}{\int }_0^x\left(\frac{1}{\sin t}-\frac{1}{t}\right)\sin \left(2nt\right)dt.$$

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Defines:

$R_n(x)$: remainder term (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

6.16.4 $$R_n(x)=O\left(n^{-1}\right),$$ $n\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

20: 36 Integrals with Coalescing Saddles

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