irregular%20singularities%20of%20rank%201

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11: Bibliography S

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K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

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B. I. Schneider, J. Segura, A. Gil, X. Guan, and K. Bartschat (2010) A new Fortran 90 program to compute regular and irregular associated Legendre functions. Comput. Phys. Comm. 181 (12), pp. 2091–2097.

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External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 8th item, §14.34(ii).
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:: Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
External Links:: ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
Referenced by:: §3.3(vi), §3.4(i), §3.5(i).
Encodings:: BibTeX

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12: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. Tables of primes (§27.21) reveal great irregularity in their distribution. … ►

Table 27.2.2: Functions related to division.

► ►►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
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: 8 Incomplete Gamma and Related
Functions

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14: 28 Mathieu Functions and Hill’s Equation

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15: 8.26 Tables

… ►

•

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

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•

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 $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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•

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

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•

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.

16: 23 Weierstrass Elliptic and Modular
Functions

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17: Bibliography W

… ►

S. S. Wagstaff (1978) The irregular primes to

125000

. Math. Comp. 32 (142), pp. 583–591.

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External Links:: FullText, Document, MathReview (Wells Johnson), ZentralBlatt
Referenced by:: §24.20.
Encodings:: BibTeX

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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R. Wong and J. F. Lin (1978) Asymptotic expansions of Fourier transforms of functions with logarithmic singularities. J. Math. Anal. Appl. 64 (1), pp. 173–180.

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External Links:: Document, MathReview (B. L. Gupta), ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

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R. Wong (1977) Asymptotic expansions of Hankel transforms of functions with logarithmic singularities. Comput. Math. Appl. 3 (4), pp. 271–286.

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External Links:: ISSN 0886-9561, Document, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

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18: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

… ►Again, there is a regular singularity at

r=0

with indices

\ell+1

and

-\ell

, and an irregular singularity of rank 1 at

r=\infty

. … ►

§33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$

… ►

33.14.7

h\left(\epsilon,\ell;r\right)=\frac{\Gamma\left(\ell+1-\kappa\right)}{\pi% \kappa^{\ell}}\left(W_{\kappa,\ell+\frac{1}{2}}\left(2r/\kappa\right)+(-1)^{% \ell}S(\epsilon,r)\frac{\Gamma\left(\ell+1+\kappa\right)}{2(2\ell+1)!}M_{% \kappa,\ell+\frac{1}{2}}\left(2r/\kappa\right)\right),

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Defines:: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\kappa$ : quantity and $S(\epsilon,r)$ : function
Referenced by:: §13.18(vi)
Permalink:: http://dlmf.nist.gov/33.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §33.14(iii), §33.14 and Ch.33

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19: 2.7 Differential Equations

… ►All other singularities are classified as irregular. … ►

§2.7(ii) Irregular Singularities of Rank 1

… ►Thus a regular singularity has rank 0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …

20: 30.2 Differential Equations

… ►This equation has regular singularities at

z=\pm 1

with exponents

\pm\frac{1}{2}\mu

and an irregular singularity of rank 1 at

z=\infty

(if

\gamma\neq 0

). … …

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
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
…