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1: 19.2 Definitions

… ►where

p_{j}

is a polynomial in

t

while

\rho

and

\sigma

are rational functions of

t

. … ►Here

a, b, p

are real parameters, and

k_{c}

and

x

are real or complex variables, with

p\neq 0

k_{c}\neq 0

. … ►If

1<k\leq 1/\sin\phi

, then

k_{c}

is pure imaginary. … ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

… ►For the special cases of

R_{C}\left(x,x\right)

and

R_{C}\left(0,y\right)

see (19.6.15). …

2: 3.1 Arithmetics and Error Measures

… ►with

b_{0}=1

and all allowable choices of

E

p

s

, and

b_{j}

. … ►Let

E_{{\rm min}}\leq E\leq E_{{\rm max}}

with

E_{{\rm min}}<0

and

E_{{\rm max}}>0

. …The integers

p

E_{{\rm min}}

, and

E_{{\rm max}}

are characteristics of the machine. … ►The current floating point arithmetic standard is IEEE 754-2019 IEEE (2019), a minor technical revision of IEEE 754-2008 IEEE (2008), which was adopted in 2011 by the International Standards Organization as ISO/IEC/IEEE 60559. … ►

N_{{\rm min}}\leq x\leq N_{{\rm max}}

, and …

3: DLMF Project News

error generating summary

4: Bibliography I

… ►

IEEE (2019) IEEE International Standard for Information Technology—Microprocessor Systems—Floating-Point arithmetic: IEEE Std 754-2019. The Institute of Electrical and Electronics Engineers, Inc..

ⓘ

Notes:: IEEE Std ISO/IEC/IEEE 60559, Revision of IEEE Std 754-1985
External Links:: ISBN 978-2-88912-566-1, Document
Referenced by:: Figure 3.1.1, Figure 3.1.1, §3.1(i), §3.1(i), Erratum (V1.0.25) for Section 3.1.
Encodings:: BibTeX

… ►

Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of

J_{0}(z)-iJ_{1}(z)

and of Bessel functions

J_{m}(z)

of any real order

m

. Linear Algebra Appl. 194, pp. 35–70.

ⓘ

External Links:: ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

… ►

Y. Ikebe (1975) The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.

ⓘ

External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

… ►

A. E. Ingham (1933) An integral which occurs in statistics. Proceedings of the Cambridge Philosophical Society 29, pp. 271–276.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §35.3(i), §35.3(ii).
Encodings:: BibTeX

… ►

M. E. H. Ismail and E. Koelink (2011) The

J

-matrix method. Adv. in Appl. Math. 46 (1-4), pp. 379–395.

ⓘ

External Links:: ISSN 0196-8858, Document, Link, MathReview (Boris Petrovich Osilenker)
Referenced by:: §18.39(iv).
Encodings:: BibTeX

…

5: Richard A. Askey

… ► 2019) was Professor Emeritus in the Department of Mathematics at the University of Wisconsin-Madison. … ►Askey was presented a Lifetime Achievement Award in Recognition and Appreciation for his Outstanding Work and Leadership in the Field of Special Functions at the International Symposium on Orthogonal Polynomials, Special Functions and Applications in Hagenberg, Austria on July 24, 2019. …

6: Bonita V. Saunders

… ►In 2019 she was named a Fellow of the Washington Academy of Sciences. …

7: Bibliography B

… ►

R. W. Barnard, K. Pearce, and K. C. Richards (2000) A monotonicity property involving

{}_{3}F_{2}

and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

ⓘ

External Links:: ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §19.9(i), §19.9(i).
Encodings:: BibTeX

… ►

E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near

T_{c}

. Phys. Rev. Lett. 31, pp. 1409–1411.

ⓘ

External Links:: Document
Referenced by:: §32.16.
Encodings:: BibTeX

… ►

F. Bethuel (1998) Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

ⓘ

External Links:: ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

… ►

W. Bühring (1987b) The behavior at unit argument of the hypergeometric function

{}_{3}F_{2}

. SIAM J. Math. Anal. 18 (5), pp. 1227–1234.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (K. M. Saksena), ZentralBlatt
Referenced by:: §16.8(ii).
Encodings:: BibTeX

… ►

P. L. Butzer and T. H. Koornwinder (2019) Josef Meixner: his life and his orthogonal polynomials. Indag. Math. (N.S.) 30 (1), pp. 250–264.

ⓘ

External Links:: ISSN 0019-3577, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

…

8: 19.11 Addition Theorems

… ►

19.11.5

\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left% (\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.6_5

R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right).

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos\NVar{z}$ : cosine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)
Permalink:: http://dlmf.nist.gov/19.11.E6_5
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.28):: This equation was given a number.
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.10

\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)-\Pi\left(\theta% ,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\gamma$ and $\delta$
Permalink:: http://dlmf.nist.gov/19.11.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

… ►

19.11.16

\Pi\left(\psi,\alpha^{2},k\right)=2\Pi\left(\theta,\alpha^{2},k\right)+\alpha^% {2}R_{C}\left(\gamma-\delta,\gamma\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\delta$ , $\psi$ : angle and $\gamma$
Permalink:: http://dlmf.nist.gov/19.11.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(iii), §19.11 and Ch.19

…

9: Bibliography N

… ►

E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

ⓘ

Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

►

E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

C. J. Noble (2004) Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

ⓘ

Notes:: Double-precision Fortran-90 code.
External Links:: Document, Code
Referenced by:: §33.23(vi), 8th item.
Encodings:: BibTeX

…

10: Errata

… ►

Version 1.0.25 (December 15, 2019)

… ►

Section 3.1

In ¶IEEE Standard (in §3.1(i)), the description was modified to reflect the most recent IEEE 754-2019 Floating-Point Arithmetic Standard IEEE (2019). In the new standard, single, double and quad floating-point precisions are replaced with new standard names of binary32, binary64 and binary128. Figure 3.1.1 has been expanded to include the binary128 floating-point memory positions and the caption has been updated using the terminology of the 2019 standard. A sentence at the end of Subsection 3.1(ii) has been added referring readers to the IEEE Standards for Interval Arithmetic IEEE (2015, 2018).

Suggested by Nicola Torracca.

… ►

Version 1.0.24 (September 15, 2019)

… ►

Version 1.0.23 (June 15, 2019)

… ►

Version 1.0.22 (March 15, 2019)

…