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1: 14.31 Other Applications

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§14.31(i) Toroidal Functions

►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

2: Bibliography H

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Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, pp. 75–115.

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Notes:

Reprinted in Ramanujan (1962, pp. 276–309).

External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

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Optical Solitons in Fibers. Springer-Verlag, Berlin, Germany.

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External Links:

ISBN 0387506683

Referenced by:

§22.19(iii).

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Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.

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External Links:

MathReview (Henrik Pedersen), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

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Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.

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External Links:

ISSN 0010-4655, Document, Code, MathReview Entry, ZentralBlatt

Referenced by:

§14.31(i).

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Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:

Reprinted in 1979.

External Links:

ISBN 0-8218-1556-3, MathReview, ZentralBlatt

Referenced by:

§35.4(i).

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3: Alexander I. Bobenko

… ► Matveev), published by Springer in 1994, Painlevé Equations in the Differential Geometry of Surfaces (with U.  Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …He is also coeditor of Discrete Integrable Geometry and Physics (with R.  Seiler), published by Oxford University Press in 1999, and Discrete Differential Geometry (with P. … Ziegler), published by Birkhäuser in 2008. …

4: Michael V. Berry

… ► 1941 in Frimley, U. … ►Berry has published numerous papers on theoretical physics, mainly in quantum mechanics and optics and including the development of associated mathematics, especially asymptotics and geometry. ►Berry has received many awards for his work in physics. …He was knighted in 1996. …National Academy of Sciences in 1995, and of the Royal Netherlands Academy of Arts and Sciences in 2000. …

5: Peter L. Walker

… ► 1942 in Dorchester, U. …He began his academic career in 1964 at the University of Lancaster, U. … ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►Walker is now retired and living in Cheltenham, UK. …

6: 26.19 Mathematical Applications

… ►Combinatorics has applications to analysis, algebra, and geometry. Examples can be found in Beckenbach (1981), Billera et al. (1996), and Lovász et al. (1995). … ►These have applications in operations research, probability theory, and statistics. …

7: Tom M. Apostol

… ► 1923 in Helper, Utah, d. … ►He received his bachelor of science in chemical engineering in 1944 and a master’s degree in mathematics in 1946, both from the University of Washington, Seattle. In 1948, he received his Ph. …In 1950, he arrived at Caltech as an assistant professor; he was named associate professor in 1956, professor in 1962, and professor emeritus in 1992. … ►In addition, he was the co-author of New Horizons in Geometry, published by the MAA, which received the CHOICE “Outstanding Academic Title” award in 2013. …

8: 36.15 Methods of Computation

… ►Close to the origin $𝐱=\mathrm{𝟎}$ of parameter space, the series in §36.8 can be used. … ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. … ►There is considerable freedom in the choice of deformations. …

9: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:

MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

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Computation of Special Functions. John Wiley & Sons Inc., New York.

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Notes:

Includes diskette containing a large collection of mathematical function software written in Fortran. Implementation in double precision. Maximum accuracy 16S.

External Links:

ISBN 0-471-11963-6, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

6th item, 3rd item, 1st item, 11st item, 7th item, 2nd item, 6th item, 3rd item, 2nd item, 5th item, 8th item, 4th item, 2nd item, §19.37(ii), §19.37(iii), §19.37(iii), §22.21, §24.19(i), §28.1, item (b), item (d), 8th item, 3rd item, 6th item, §5.22(ii), §5.22(iii), 2nd item, 2nd item, 4th item, 2nd item, 2nd item, §8.17(v), 4th item, 2nd item, 5th item, 3rd item, 3rd item, 3rd item, § ‣ 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, § ‣ 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.

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A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

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Notes:

English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905

External Links:

ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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10: 22.18 Mathematical Applications

… ►The arc length $l(u)$ in the first quadrant, measured from $u=0$, is … ►Bowman (1953, Chapters V–VI) gives an overview of the use of Jacobian elliptic functions in conformal maps for engineering applications. … ►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. …Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4). … ►The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973). …