change%20of%20order%20of%20integration

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21: 27.15 Chinese Remainder Theorem

… ►Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

22: William P. Reinhardt

… ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

23: 9.7 Asymptotic Expansions

… ►

9.7.1

\zeta=\tfrac{2}{3}z^{\ifrac{3}{2}}.

ⓘ

Defines:: $\zeta$ : change of variable (locally)
Symbols:: $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(i), §9.7 and Ch.9

… ►Numerical values of

\chi(n)

are given in Table 9.7.1 for

n=1(1)20

to 2D. ►

Table 9.7.1:

\chi(n)

► ►►►

$n$	$\chi(n)$	$n$	$\chi(n)$	$n$	$\chi(n)$	$n$	$\chi(n)$
…
5	2.95	10	4.06	15	4.94	20	5.68

► … ►

9.7.20

R_{n}(z)=(-1)^{n}\sum_{k=0}^{m-1}(-1)^{k}u_{k}\frac{G_{n-k}\left(2\zeta\right)% }{\zeta^{k}}+R_{m,n}(z),

ⓘ

Defines:: $R_{n}$ : remainder function (locally)
Symbols:: $G_{\NVar{p}}\left(\NVar{z}\right)$ : rescaled terminant function, $k$ : nonnegative integer, $z$ : complex variable, $\zeta$ : change of variable, $m$ : index, $n$ : index and $u_{s}$ : expansion coefficient
Proof sketch:: Use the methods of Olver (1991b, 1993a).
Permalink:: http://dlmf.nist.gov/9.7.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(v), §9.7 and Ch.9

… ►

9.7.23

R_{m,n}(z),S_{m,n}(z)=O\left(e^{-2|\zeta|}\zeta^{-m}\right),

|\operatorname{ph}z|\leq\tfrac{2}{3}\pi

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $z$ : complex variable, $\zeta$ : change of variable, $m$ : index, $n$ : index, $R_{n}$ : remainder function and $S_{n}$ : remainder function
Proof sketch:: Use the methods of Olver (1991b, 1993a).
Permalink:: http://dlmf.nist.gov/9.7.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(v), §9.7 and Ch.9

…

24: 20.11 Generalizations and Analogs

… ►

§20.11(iii) Ramanujan’s Change of Base

… ►These results are called Ramanujan’s changes of base. …

Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

… ►

•

Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

►

•

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.