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11: 33.13 Complex Variable and Parameters

… ►The quantities

C_{\ell}\left(\eta\right)

{\sigma_{\ell}}\left(\eta\right)

, and

R_{\ell}

, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that ►

33.13.1

C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, ${\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)$ : Coulomb phase shift, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ell$ : nonnegative integer and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.13 and Ch.33

… ►

33.13.2

R_{\ell}=(2\ell+1)C_{\ell}\left(\eta\right)/C_{\ell-1}\left(\eta\right).

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\ell$ : nonnegative integer, $\eta$ : real parameter and $R_{\ell}$ : factor
Permalink:: http://dlmf.nist.gov/33.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §33.13 and Ch.33

►For further information see Dzieciol et al. (1999), Thompson and Barnett (1986), and Humblet (1984).

12: Richard A. Askey

… ► Roy), a standard text on the topic, was published by Cambridge University Press in 1999. … ► C. … ►National Academy of Sciences in 1999. …

13: 15.11 Riemann’s Differential Equation

… ►

15.11.2

a_{1}+a_{2}+b_{1}+b_{2}+c_{1}+c_{2}=1.

ⓘ

Symbols:: $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

►Here

\{a_{1},a_{2}\}

\{b_{1},b_{2}\}

\{c_{1},c_{2}\}

are the exponent pairs at the points

\alpha

\beta

\gamma

, respectively. … ►

15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►

15.11.4

w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►

15.11.6

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

…

… ► Nayfeh and C. … ►Clark was elected a Fellow of the American Physical Society (APS) in 1992, of the Optical Society of America (OSA) in 1994, of the Institute of Physics in 1999, of the American Association for the Advancement of Science (AAAS) in 2001, and of the Washington Academy of Sciences in 2003. …

15: 15.2 Definitions and Analytical Properties

… ►In general,

F\left(a,b;c;z\right)

does not exist when

c=0,-1,-2,\dots

. … ►For all values of

c

… ►

(c)

Diverges when $\Re\left(c-a-b\right)\leq-1$ .

… ►The principal branch of

\mathbf{F}\left(a,b;c;z\right)

is an entire function of

a

b

, and

c

. …The same properties hold for

F\left(a,b;c;z\right)

, except that as a function of

c

F\left(a,b;c;z\right)

in general has poles at

c=0,-1,-2,\dots

. …

16: Bibliography K

… ►

K. Kajiwara and T. Masuda (1999) On the Umemura polynomials for the Painlevé III equation. Phys. Lett. A 260 (6), pp. 462–467.

ⓘ

External Links:: ISSN 0375-9601, MathReview, ZentralBlatt
Referenced by:: §32.8(iii).
Encodings:: BibTeX

… ►

E. G. Kalnins, W. Miller, G. F. Torres del Castillo, and G. C. Williams (2000) Special Functions and Perturbations of Black Holes. In Special Functions (Hong Kong, 1999), pp. 140–151.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

… ►

M. K. Kerimov (1999) The Rayleigh function: Theory and computational methods. Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.

ⓘ

Notes:: Translation in Comput. Math. Math. Phys. 39 (1999), No. 12, pp. 1883–1925.
External Links:: ISSN 0044-4669, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.21(xiii).
Encodings:: BibTeX

… ►

N. Koblitz (1999) Algebraic Aspects of Cryptography. Springer-Verlag, Berlin.

ⓘ

Notes:: Second printing with corrections
External Links:: ISBN 3-540-63446-0, MathReview (M. V. D. Burmester), ZentralBlatt
Referenced by:: §23.20(iii).
Encodings:: BibTeX

… ►

T. Kriecherbauer and K. T.-R. McLaughlin (1999) Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.

ⓘ

External Links:: ISSN 1073-7928, Document, MathReview (Heinrich Begehr), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

…

17: 26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). …

18: 26 Combinatorial Analysis

…

19: 25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …

20: Bibliography N

… ►

M. Neher (2007) Complex standard functions and their implementation in the CoStLy library. ACM Trans. Math. Softw. 33 (1), pp. Article 2.

ⓘ

Notes:: For the software see CoStLy (free C-XSC library)
External Links:: ISSN 0098-3500, Document
Referenced by:: 3rd item, CoStLy (free C-XSC library).
Encodings:: BibTeX

… ►

P. Nevai (1986) Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

M. Noumi and Y. Yamada (1999) Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, pp. 53–86.

ⓘ

External Links:: ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

… ►

Numerical Recipes (commercial C, C++, Fortran 77, and Fortran 90 libraries)

ⓘ

Notes:: A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

J. F. Nye (1999) Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. Institute of Physics Publishing, Bristol.

ⓘ

External Links:: ISBN 0-7503-0610-6, MathReview (Peter Giblin), ZentralBlatt
Referenced by:: §36.14(ii).
Encodings:: BibTeX

…