DLMF: Untitled Document

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Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\operatorname{erf}x$ on $0\leq x\leq 2$ , for $xe^{x^{2}}\operatorname{erfc}x$ on $[2,\infty)$ , and for $e^{x^{2}}\operatorname{erfc}x$ on $[0,\infty)$ (30D).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x)e^{x^{2}}\operatorname{erfc}x$ on $(0,\infty)$ (22D).

…

… ►It counts the number of lattice paths from

(0,0)

to

(n,n)

that stay on or above the line

y=x

. … ►

Table 26.5.1: Catalan numbers.

► ►►►

$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
…
6	132	13	7 42900	20	65641 20420

► …

… ►

\genfrac{(}{)}{0.0pt}{}{m+n}{n}

is the number of lattice paths from

(0,0)

to

(m,n)

. …The number of lattice paths from

(0,0)

to

(m,n)

,

m\leq n

, that stay on or above the line

y=x

is

\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.

… ►

Table 26.3.1: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

.

► ►►►

$m$	$n$
$m$	…
6	1	6	15	20	15	6	1
…

► ►

Table 26.3.2: Binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m+n}{m}

for lattice paths.

► ►►►

$m$	$n$
$m$	…
3	1	4	10	20	35	56	84	120	165
…

► …

… ►Unless otherwise specified, it consists of horizontal segments corresponding to the vector

(1,0)

and vertical segments corresponding to the vector

(0,1)

. … ►

Table 26.2.1: Partitions

p\left(n\right)

.

► ►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
3	3	20	627	37	21637
…

►

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•

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.

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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.

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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.

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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.

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\mathrm{Me}_{n}^{(1,2)}(z,q)=\sqrt{\tfrac{1}{2}\pi}g_{\mathit{e},n}(h)% \operatorname{ce}_{n}\left(0,q\right){\operatorname{Mc}^{(3,4)}_{n}}\left(z,h% \right),

… ►

\mathrm{Ne}_{n}^{(1,2)}(z,q)=\sqrt{\tfrac{1}{2}\pi}g_{\mathit{o},n}(h)% \operatorname{se}_{n}'\left(0,q\right){\operatorname{Ms}^{(3,4)}_{n}}\left(z,h% \right).

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Abramowitz and Stegun (1964, Chapter 20)

…

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N. W. Macfadyen and P. Winternitz (1971) Crossing symmetric expansions of physical scattering amplitudes: The

O(2,1)

group and Lamé functions. J. Mathematical Phys. 12, pp. 281–293.

ⓘ

External Links:: Document, MathReview
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

S. M. Markov (1981) On the interval computation of elementary functions. C. R. Acad. Bulgare Sci. 34 (3), pp. 319–322.

ⓘ

External Links:: ISSN 0366-8681, MathReview (David F. Griffiths), ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

…

… ►If

f(a)f(b)<0

with

a<b

, then the interval

[a,b]

contains one or more zeros of

f

. …All zeros of

f

in the original interval

[a,b]

can be computed to any predetermined accuracy. … ►The convergence is linear, and again more than one zero may occur in the original interval

[x_{0},x_{1}]

. … ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. …

… ►

R. B. Paris (2005b) The Stokes phenomenon associated with the Hurwitz zeta function

\zeta(s,a)

. Proc. Roy. Soc. London Ser. A 461, pp. 297–304.

ⓘ

External Links:: ISSN 1364-5021, Document, MathReview (Marc Prevost), ZentralBlatt
Referenced by:: (25.11.43), §25.11(xii).
Encodings:: BibTeX

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M. S. Petković and L. D. Petković (1998) Complex Interval Arithmetic and its Applications. Mathematical Research, Vol. 105, Wiley-VCH Verlag Berlin GmbH, Berlin.

ⓘ

External Links:: ISBN 3-527-40134-2, MathReview (G. Alefeld), ZentralBlatt
Referenced by:: §3.1(ii).
Encodings:: BibTeX

… ►

R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

…

… ►

Table 26.9.1: Partitions

p_{k}\left(n\right)

.

► ►►►

$n$	$k$
$n$	…
8	0	1	5	10	15	18	20	21	22	22	22
…

► … ►It is also equal to the number of lattice paths from

(0,0)

to

(m,k)

that have exactly

n

vertices

(h,j)

,

1\leq h\leq m

,

1\leq j\leq k

, above and to the left of the lattice path. …

on%20intervals

11—20 of 30 matching pages

11: 7.24 Approximations

12: 26.5 Lattice Paths: Catalan Numbers

13: 26.3 Lattice Paths: Binomial Coefficients

14: 26.2 Basic Definitions

15: 9.18 Tables

16: 28.1 Special Notation

Abramowitz and Stegun (1964, Chapter 20)

17: Bibliography M

18: 3.8 Nonlinear Equations

19: Bibliography P

20: 26.9 Integer Partitions: Restricted Number and Part Size