DLMF: Untitled Document

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B. Hall (2015) Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:: An elementary introduction
External Links:: ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry
Referenced by:: §1.2(vi).
Encodings:: BibTeX

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B. A. Hargrave (1978) High frequency solutions of the delta wing equations. Proc. Roy. Soc. Edinburgh Sect. A 81 (3-4), pp. 299–316.

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External Links:: ISSN 0308-2105, MathReview (V. M. Babich), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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G. J. Heckman (1991) An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. 103 (2), pp. 341–350.

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External Links:: ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

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M. Heil (1995) Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems. Ph.D. Thesis, Technischen Universität Berlin.

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Notes:: (All relevant material, plus corrections, are included in Deconinck et al. (2004).)
External Links:: ZentralBlatt
Referenced by:: §21.5(i).
Encodings:: BibTeX

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P. W. Hemker, T. H. Koornwinder, and N. M. Temme (1993) Wavelets: mathematical preliminaries. In Wavelets: an elementary treatment of theory and applications, Ser. Approx. Decompos., Vol. 1, pp. 13–26.

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External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Tom H. Koornwinder.
Encodings:: BibTeX

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… ►Her research interests include numerical grid generation, numerical solution of partial differential equations, and visualization of special functions. … ►As the principal developer of graphics for the DLMF, she has collaborated with other NIST mathematicians, computer scientists, and student interns to produce informative graphs and dynamic interactive visualizations of elementary and higher mathematical functions over both simply and multiply connected domains. …

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§28.28(i) Equations with Elementary Kernels

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28.28.1

w=\cosh z\cos t\cos\alpha+\sinh z\sin t\sin\alpha.

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Symbols:: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/28.28.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(i), §28.28 and Ch.28

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28.28.2

\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\operatorname{ce}_{n}\left(t,h^% {2}\right)\,\mathrm{d}t={\mathrm{i}}^{n}\operatorname{ce}_{n}\left(\alpha,h^{2% }\right){\operatorname{Mc}^{(1)}_{n}}\left(z,h\right),

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $h$ : parameter, $n$ : integer, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $\alpha$ : parameter
A&S Ref:: 20.7.34 20.7.35 20.7.36 20.7.37
Permalink:: http://dlmf.nist.gov/28.28.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(i), §28.28 and Ch.28

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28.28.3

\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\operatorname{se}_{n}\left(t,h^% {2}\right)\,\mathrm{d}t={\mathrm{i}}^{n}\operatorname{se}_{n}\left(\alpha,h^{2% }\right){\operatorname{Ms}^{(1)}_{n}}\left(z,h\right),

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $h$ : parameter, $n$ : integer, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/28.28.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(i), §28.28 and Ch.28

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28.28.4

\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{\partial w}{\partial\alpha}e^{2% \mathrm{i}hw}\operatorname{ce}_{n}\left(t,h^{2}\right)\,\mathrm{d}t={\mathrm{i% }}^{n}\operatorname{ce}_{n}'\left(\alpha,h^{2}\right){\operatorname{Mc}^{(1)}_% {n}}\left(z,h\right),

…

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NAG (commercial C and Fortran libraries) Numerical Algorithms Group, Ltd..

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Notes:: The NAG Libraries are large general purpose numerical software libraries with broad coverage of elementary and special functions. Implementations are in single and double precision.
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, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:: ISSN 0031-9015, Document, MathReview, ZentralBlatt
Referenced by:: §18.38(ii).
Encodings:: BibTeX

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J. J. Nestor (1984) Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:: §2.8(vi).
Encodings:: BibTeX

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L. N. Nosova and S. A. Tumarkin (1965) Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations

\epsilon(py^{\prime})^{\prime}+(q+\epsilon r)y=f

. Pergamon Press, Oxford.

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Notes:: D. E. Brown, translator.
External Links:: MathReview, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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External Links:: MathReview (E. A. Boyd), ZentralBlatt
Referenced by:: (9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).
Encodings:: BibTeX

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W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

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W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

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G. M. Roper (1951) Some Applications of the Lamé Function Solutions of the Linearised Supersonic Flow Equations. Technical Reports and Memoranda Technical Report 2865, Aeronautical Research Council (Great Britain).

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Notes:: Reprinted by Her Majesty’s Stationery Office, London, 1962.
External Links:: MathReview (P. Germain)
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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K. H. Rosen (2004) Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.

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External Links:: ISBN 0-201-87073-8, MathReview (Norman J. Richert)
Referenced by:: §27.17.
Encodings:: BibTeX

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§12.14(i) Introduction

►In this section solutions of equation (12.2.3) are considered. … ►Here

w_{1}(a,x)

and

w_{2}(a,x)

are the even and odd solutions of (12.2.3): … ►The even and odd solutions of (12.2.3) (see §12.14(v)) are given by … ►

Positive $a$ , $2\sqrt{a}<x<\infty$

…

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T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

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

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

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X. Merrheim (1994) The computation of elementary functions in radix

2^{p}

. Computing 53 (3-4), pp. 219–232.

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Notes:: International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Vienna, 1993)
External Links:: ISSN 0010-485X, Document, MathReview, ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

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S. C. Milne (1985b) An elementary proof of the Macdonald identities for

A_{l}^{(1)}

. Adv. in Math. 57 (1), pp. 34–70.

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External Links:: ISSN 0001-8708, Document, MathReview (Y. Lehrer-Ilamed), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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J. Muller (1997) Elementary Functions: Algorithms and Implementation. Birkhäuser Boston Inc., Boston, MA.

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Notes:: A 2nd edition was published in 2006.
External Links:: ISBN 0-8176-3990-X, MathReview (Warren E. Ferguson, Jr.), ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

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W. Kahan (1987) Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit. In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9, pp. 165–211.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §4.23(iv), §4.37(iv).
Encodings:: BibTeX

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K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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External Links:: ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

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K. Kajiwara and Y. Ohta (1998) Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.

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External Links:: ISSN 0305-4470, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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A. A. Kapaev (1988) Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:: In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)
External Links:: ISSN 0374-0641, MathReview, ZentralBlatt
Referenced by:: §32.11(i).
Encodings:: BibTeX

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Y. A. Kravtsov (1964) Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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Referenced by:: §36.14(i).
Encodings:: BibTeX

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… ►For examples of minimax polynomial approximations to elementary and special functions see Hart et al. (1968). … ►There exists a unique solution of this minimax problem and there are at least

k+\ell+2

values

x_{j}

,

a\leq x_{0}<x_{1}<\cdots<x_{k+\ell+1}\leq b

, such that

m_{j}=m

, where … ►A collection of minimax rational approximations to elementary and special functions can be found in Hart et al. (1968). … ►With

b_{0}=1

, the last

q

equations give

b_{1},\dots,b_{q}

as the solution of a system of linear equations. … ►If the functions

\phi_{k}(x)

are linearly independent on the set

x_{1},x_{2},\dots,x_{J}

, that is, the only solution of the system of equations …

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A. Cayley (1895) An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.

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Notes:: Republished by Dover Publications, Inc,, New York, 1961. Table erratum: Math. Comp. v. 29 (1975), no. 130, p. 670.
External Links:: ZentralBlatt
Referenced by:: §19.12, Ch.19, A. Cayley (1961).
Encodings:: BibTeX

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A. Cayley (1961) An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).

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Notes:: Second edition of Cayley (1895), unabridged and corrected
External Links:: ZentralBlatt
Referenced by:: §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i).
Encodings:: BibTeX

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T. W. Chaundy (1969) Elementary Differential Equations. Clarendon Press, Oxford.

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External Links:: ISBN 0-19-853139-7, MathReview, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

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W. J. Cody and W. Waite (1980) Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs.

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External Links:: ISBN 0138220646, ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

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W. J. Cody (1993a) Algorithm 714: CELEFUNT – A portable test package for complex elementary functions. ACM Trans. Math. Software 19 (1), pp. 1–21.

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External Links:: ISSN 0098-3500, Document, GAMS (module 714; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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elementary solutions

11—20 of 32 matching pages

11: Bibliography H

12: Bonita V. Saunders

13: 28.28 Integrals, Integral Representations, and Integral Equations

§28.28(i) Equations with Elementary Kernels

14: Bibliography N

15: Bibliography R

16: 12.14 The Function $W\left(a,x\right)$

§12.14(i) Introduction

Positive $a$ , $2\sqrt{a}<x<\infty$

17: Bibliography M

18: Bibliography K

19: 3.11 Approximation Techniques

20: Bibliography C

elementary solutions

11—20 of 32 matching pages

11: Bibliography H

12: Bonita V. Saunders

13: 28.28 Integrals, Integral Representations, and Integral Equations

§28.28(i) Equations with Elementary Kernels

14: Bibliography N

15: Bibliography R

16: 12.14 The Function W ⁡ ( a , x )

§12.14(i) Introduction

Positive a , 2 ⁢ a < x < ∞

17: Bibliography M

18: Bibliography K

19: 3.11 Approximation Techniques

20: Bibliography C

16: 12.14 The Function $W\left(a,x\right)$

Positive $a$ , $2\sqrt{a}<x<\infty$