Atjaunināt sīkdatņu piekrišanu

Special Functions and Orthogonal Polynomials [Hardback]

(City University of Hong Kong), (Yale University, Connecticut)
  • Formāts: Hardback, 488 pages, height x width x depth: 237x154x32 mm, weight: 780 g, Worked examples or Exercises; 7 Line drawings, unspecified
  • Sērija : Cambridge Studies in Advanced Mathematics
  • Izdošanas datums: 17-May-2016
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1107106982
  • ISBN-13: 9781107106987
Citas grāmatas par šo tēmu:
  • Hardback
  • Cena: 100,23 €
  • Grāmatu piegādes laiks ir 3-4 nedēļas, ja grāmata ir uz vietas izdevniecības noliktavā. Ja izdevējam nepieciešams publicēt jaunu tirāžu, grāmatas piegāde var aizkavēties.
  • Daudzums:
  • Ielikt grozā
  • Piegādes laiks - 4-6 nedēļas
  • Pievienot vēlmju sarakstam
Special Functions and Orthogonal PolynomialsPalielināt
  • Formāts: Hardback, 488 pages, height x width x depth: 237x154x32 mm, weight: 780 g, Worked examples or Exercises; 7 Line drawings, unspecified
  • Sērija : Cambridge Studies in Advanced Mathematics
  • Izdošanas datums: 17-May-2016
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 1107106982
  • ISBN-13: 9781107106987
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781107106987.html
Keywords:
The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painlevé transcendents, which have been termed the 'special functions of the twenty-first century'.

Recenzijas

' an excellent graduate textbook, one of the two best available on this subject' Warren Johnson, MAA Reviews (www.maa.org)

Papildus informācija

A comprehensive graduate-level introduction to classical and contemporary aspects of special functions.
Preface xi
1 Orientation
1(14)
1.1 Power series solutions
2(3)
1.2 The gamma and beta functions
5(1)
1.3 Three questions
6(4)
1.4 Other special functions
10(1)
1.5 Exercises
11(2)
1.6 Remarks
13(2)
2 Gamma, beta, zeta
15(31)
2.1 The gamma and beta functions
16(3)
2.2 Euler's product and reflection formulas
19(3)
2.3 Formulas of Legendre and Gauss
22(2)
2.4 Two characterizations of the gamma function
24(2)
2.5 Asymptotics of the gamma function
26(4)
2.6 The psi function and the incomplete gamma function
30(2)
2.7 The Selberg integral
32(3)
2.8 The zeta function
35(3)
2.9 Exercises
38(7)
2.10 Remarks
45(1)
3 Second-order differential equations
46(27)
3.1 Transformations and symmetry
47(2)
3.2 Existence and uniqueness
49(3)
3.3 Wronskians, Green's functions, and comparison
52(3)
3.4 Polynomials as eigenfunctions
55(5)
3.5 Maxima, minima, and estimates
60(2)
3.6 Some equations of mathematical physics
62(4)
3.7 Equations and transformations
66(2)
3.8 Exercises
68(3)
3.9 Remarks
71(2)
4 Orthogonal polynomials on an interval
73(21)
4.1 Weight functions and orthogonality
74(4)
4.2 Stieltjes transform and Pade approximants
78(3)
4.3 Pade approximants and continued fractions
81(3)
4.4 Generalization: measures
84(2)
4.5 Favard's theorem and the moment problem
86(3)
4.6 Asymptotic distribution of zeros
89(1)
4.7 Exercises
90(3)
4.8 Remarks
93(1)
5 The classical orthogonal polynomials
94(46)
5.1 Classical polynomials: general properties, I
94(4)
5.2 Classical polynomials: general properties, II
98(4)
5.3 Hermite polynomials
102(6)
5.4 Laguerre polynomials
108(3)
5.5 Jacobi polynomials
111(4)
5.6 Legendre and Chebyshev polynomials
115(5)
5.7 Distribution of zeros and electrostatics
120(4)
5.8 Expansion theorems
124(6)
5.9 Functions of the second kind
130(3)
5.10 Exercises
133(4)
5.11 Remarks
137(3)
6 Semi-classical orthogonal polynomials
140(32)
6.1 Discrete weights and difference operators
141(5)
6.2 The discrete Rodrigues formula
146(3)
6.3 Charlier polynomials
149(3)
6.4 Krawtchouk polynomials
152(3)
6.5 Meixner polynomials
155(3)
6.6 Chebyshev-Hahn polynomials
158(4)
6.7 Neo-classical polynomials
162(6)
6.8 Exercises
168(2)
6.9 Remarks
170(2)
7 Asymptotics of orthogonal polynomials: two methods
172(28)
7.1 Approximation away from the real line
173(2)
7.2 Asymptotics by matching
175(3)
7.3 The Riemann-Hilbert formulation
178(1)
7.4 The Riemann-Hilbert problem in the Hermite case, I
179(6)
7.5 The Riemann-Hilbert problem in the Hermite case, II
185(7)
7.6 Hermite asymptotics
192(4)
7.7 Exercises
196(2)
7.8 Remarks
198(2)
8 Confluent hypergeometric functions
200(23)
8.1 Kummer functions
201(3)
8.2 Kummer functions of the second kind
204(3)
8.3 Solutions when c is an integer
207(1)
8.4 Special cases
208(2)
8.5 Contiguous functions
210(2)
8.6 Parabolic cylinder functions
212(4)
8.7 Whittaker functions
216(3)
8.8 Exercises
219(2)
8.9 Remarks
221(2)
9 Cylinder functions
223(31)
9.1 Bessel functions
224(4)
9.2 Zeros of real cylinder functions
228(4)
9.3 Integral representations
232(2)
9.4 Hankel functions
234(4)
9.5 Modified Bessel functions
238(1)
9.6 Addition theorems
239(2)
9.7 Fourier transform and Hankel transform
241(2)
9.8 Integrals of Bessel functions
243(2)
9.9 Airy functions
245(3)
9.10 Exercises
248(5)
9.11 Remarks
253(1)
10 Hypergeometric functions
254(25)
10.1 Solutions of the hypergeometric equation
255(3)
10.2 Linear relations of solutions
258(3)
10.3 Solutions when c is an integer
261(2)
10.4 Contiguous functions
263(3)
10.5 Quadratic transformations
266(3)
10.6 Integral transformations and special values
269(4)
10.7 Exercises
273(4)
10.8 Remarks
277(2)
11 Spherical functions
279(26)
11.1 Harmonic polynomials and surface harmonics
280(6)
11.2 Legendre functions
286(3)
11.3 Relations among the Legendre functions
289(4)
11.4 Series expansions and asymptotics
293(3)
11.5 Associated Legendre functions
296(3)
11.6 Relations among associated functions
299(2)
11.7 Exercises
301(2)
11.8 Remarks
303(2)
12 Generalized hypergeometric functions; G-functions
305(25)
12.1 Generalized hypergeometric series
305(3)
12.2 The generalized hypergeometric equation
308(4)
12.3 Meijer G-functions
312(7)
12.4 Choices of contour of integration
319(3)
12.5 Expansions and asymptotics
322(3)
12.6 The Mellin transform and G-functions
325(1)
12.7 Exercises
326(2)
12.8 Remarks
328(2)
13 Asymptotics
330(28)
13.1 Hermite and parabolic cylinder functions
331(2)
13.2 Confluent hypergeometric functions
333(5)
13.3 Hypergeometric functions and Jacobi polynomials
338(2)
13.4 Legendre functions
340(2)
13.5 Steepest descents and stationary phase
342(3)
13.6 Exercises
345(12)
13.7 Remarks
357(1)
14 Elliptic functions
358(32)
14.1 Integration
359(2)
14.2 Elliptic integrals
361(5)
14.3 Jacobi elliptic functions
366(5)
14.4 Theta functions
371(4)
14.5 Jacobi theta functions and integration
375(5)
14.6 Weierstrass elliptic functions
380(3)
14.7 Exercises
383(5)
14.8 Remarks
388(2)
15 Painleve transcendents
390(40)
15.1 The Painleve method
392(4)
15.2 Derivation of PII
396(3)
15.3 Solutions of PII
399(3)
15.4 Compatibility conditions and Backlund transformations
402(6)
15.5 Construction of ψ
408(4)
15.6 Monodromy and isomonodromy
412(3)
15.7 The inverse problem and the Painleve property
415(4)
15.8 Asymptotics of PII(0)
419(5)
15.9 Exercises
424(4)
15.10 Remarks
428(2)
Appendix A Complex analysis
430(7)
A.1 Holomorphic and meromorphic functions
430(1)
A.2 Cauchy's theorem, the Cauchy integral theorem, and Liouville's theorem
431(1)
A.3 The residue theorem and counting zeros
432(2)
A.4 Linear fractional transformations
434(1)
A.5 Weierstrass factorization theorem
434(1)
A.6 Cauchy and Stieltjes transformations and the Sokhotski--Plemelj formula
435(2)
Appendix B Fourier analysis
437(6)
B.1 Fourier and inverse Fourier transforms
437(1)
B.2 Proof of Theorem 4.1.5
438(1)
B.3 Riemann--Lebesgue lemma
439(1)
B.4 Fourier series and the Weierstrass approximation theorem
440(1)
B.5 The Mellin transform and its inverse
441(2)
References 443(20)
Author index 463(5)
Notation index 468(1)
Subject index 469
Richard Beals is a former Professor of Mathematics at the University of Chicago and Yale University. He is the author or co-author of books on mathematical analysis, linear operators and inverse scattering theory, and has authored more than 100 research papers in areas including partial differential equations, mathematical economics and mathematical psychology. Roderick Wong is Chair Professor of Mathematics at the City University of Hong Kong. He is the author of books on asymptotic approximations of integrals and applied analysis. He has published over 140 research papers in areas such as asymptotic analysis, singular perturbation theory and special functions.