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21: 6.20 Approximations

… ►

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

►

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

… ►

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Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\operatorname{Ein}\left(z\right)$ , $\operatorname{Si}\left(z\right)$ , $\operatorname{Cin}\left(z\right)$ (valid near the origin), and $E_{1}\left(z\right)$ (valid for large $|z|$ ); approximate errors are given for a selection of $z$ -values.

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•

Luke (1969b, pp. 411–414) gives rational approximations for $\operatorname{Ein}\left(z\right)$ .

22: 31.18 Methods of Computation

… ►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of

z

; see Laĭ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …

23: 12.11 Zeros

… ►If

a>-\tfrac{1}{2}

, then

V\left(a,x\right)

has no positive real zeros, and if

a=\tfrac{3}{2}-2n

n\in\mathbb{Z}

, then

V\left(a,x\right)

has a zero at

x=0

. … ►When

a>-\frac{1}{2}

U\left(a,z\right)

has a string of complex zeros that approaches the ray

\operatorname{ph}z=\frac{3}{4}\pi

z\to\infty

, and a conjugate string. When

a>-\frac{1}{2}

the zeros are asymptotically given by

z_{a,s}

and

\overline{z_{a,s}}

, where

s

is a large positive integer and … ►Numerical calculations in this case show that

z_{\frac{1}{2},s}

corresponds to the

s

th zero on the string; compare §7.13(ii). … ►For large negative values of

a

the real zeros of

U\left(a,x\right)

U'\left(a,x\right)

V\left(a,x\right)

, and

V'\left(a,x\right)

can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

24: 4.31 Special Values and Limits

§4.31 Special Values and Limits

►

Table 4.31.1: Hyperbolic functions: values at multiples of

\tfrac{1}{2}\pi i

► ►►

$z$	0	$\frac{1}{2}\pi\mathrm{i}$	$\pi\mathrm{i}$	$\frac{3}{2}\pi\mathrm{i}$	$\infty$
…

► ►

4.31.1

\lim_{z\to 0}\frac{\sinh z}{z}=1,

ⓘ

Symbols:: $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

►

4.31.2

\lim_{z\to 0}\frac{\tanh z}{z}=1,

ⓘ

Symbols:: $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

►

4.31.3

\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.31 and Ch.4

25: Gergő Nemes

… ► 1988 in Szeged, Hungary) is a Research Fellow at the Alfréd Rényi Institute of Mathematics in Budapest, Hungary. … ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

26: Peter L. Walker

… ►was Professor of Mathematics at the American University of Sharjah, Sharjah, United Arab Emirates, in 1997–2005. He began his academic career in 1964 at the University of Lancaster, U. …Since 1984 he has also taught at other Persian Gulf universities, including Sultan Qaboos University, Oman. … ►

20 Theta Functions

…

27: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of

g_{k}

up to

g_{20}

. Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of

g_{k}

for

k=21,22,\dots,30

. … ►uniformly for bounded real values of

x

. … ►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. …

28: 4.46 Tables

… ►Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). … ►For 40D values of the first 500 roots of

\tan x=x

, see Robinson (1972). (These roots are zeros of the Bessel function

J_{3/2}\left(x\right)

; see §10.21.) ►For 10S values of the first five complex roots of

\sin z=az

\cos z=az

, and

\cosh z=az

, for selected positive values of

a

, see Fettis (1976). …

29: 9.18 Tables

… ►

•

Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

… ►

•

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

… ►

•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

►

•

Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

…

30: 5.21 Methods of Computation

… ►An effective way of computing

\Gamma\left(z\right)

in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). Or we can use forward recurrence, with an initial value obtained e. … ►Similarly for

\ln\Gamma\left(z\right)

\psi\left(z\right)

, and the polygamma functions. …