special%20cases

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11: 32.8 Rational Solutions

… ►Special rational solutions of

\mbox{P}_{\mbox{\scriptsize III}}

are … ►Special rational solutions of

\mbox{P}_{\mbox{\scriptsize IV}}

are … ►The rational solutions when the parameters satisfy (32.8.22) are special cases of §32.10(iv). … ►Cases (a) and (b) are special cases of §32.10(v). … ►These are special cases of §32.10(vi). …

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

13: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of

g_{k}

up to

g_{20}

. … ►In the case

K=1

the factor

1+\zeta\left(K\right)

is replaced with 4. … ►For the error term in (5.11.19) in the case

z=x\;(>0)

and

c=1

, see Olver (1995). …

14: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

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
…

► … ►In the present chapter

m\geq n\geq 0

in all cases. … ►equivalently, partitions into at most

k

parts either have exactly

k

parts, in which case we can subtract one from each part, or they have strictly fewer than

k

parts. …

15: Bibliography M

… ►

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

… ►

A. J. MacLeod (2002a) Asymptotic expansions for the zeros of certain special functions. J. Comput. Appl. Math. 145 (2), pp. 261–267.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.70, §11.4(vii), §6.13.
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

… ►

R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:: FullText
Referenced by:: §8.24(i).
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

…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…