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Explorations in Complex Functions 2020 ed. [Hardback]

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  • Formāts: Hardback, 353 pages, height x width: 235x155 mm, weight: 790 g, 29 Illustrations, color; 1 Illustrations, black and white; XVI, 353 p. 30 illus., 29 illus. in color., 1 Hardback
  • Sērija : Graduate Texts in Mathematics 287
  • Izdošanas datums: 20-Oct-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030545326
  • ISBN-13: 9783030545321
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Explorations in Complex Functions 2020 ed.Palielināt
  • Formāts: Hardback, 353 pages, height x width: 235x155 mm, weight: 790 g, 29 Illustrations, color; 1 Illustrations, black and white; XVI, 353 p. 30 illus., 29 illus. in color., 1 Hardback
  • Sērija : Graduate Texts in Mathematics 287
  • Izdošanas datums: 20-Oct-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030545326
  • ISBN-13: 9783030545321
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783030545321.html
Keywords:

This textbook explores a selection of topics in complex analysis. From core material in the mainstream of complex analysis itself, to tools that are widely used in other areas of mathematics, this versatile compilation offers a selection of many different paths. Readers interested in complex analysis will appreciate the unique combination of topics and connections collected in this book.

Beginning with a review of the main tools of complex analysis, harmonic analysis, and functional analysis, the authors go on to present multiple different, self-contained avenues to proceed. Chapters on linear fractional transformations, harmonic functions, and elliptic functions offer pathways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions lead into L-functions, while a chapter on entire functions opens pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give rise to Hilbert and Fourier transforms, with an emphasis on the connection to complex analysis. Valuable additional topics include Riemann surfaces, steepest descent, tauberian theorems, and the Wiener–Hopf method.

Showcasing an array of accessible excursions, Explorations in Complex Functions is an ideal companion for graduate students and researchers in analysis and number theory. Instructors will appreciate the many options for constructing a second course in complex analysis that builds on a first course prerequisite; exercises complement the results throughout.


Recenzijas

This is a suitable book with a proper concept at the right time. It is suitable because it shows the beauty, power and profundity of complex analysis, enlightens how many sided it is with all its inspirations and cross-connections to other branches of mathematics. (Heinrich Begehr, zbMATH 1460.30001, 2021)

1 Basics
1(20)
1.1 The Cauchy-Riemann equations and Cauchy's integral theorem
1(2)
1.2 The Cauchy integral formula and applications
3(3)
1.3 Change of contour, isolated singularities, residues
6(3)
1.4 The logarithm and powers
9(1)
1.5 Infinite products
10(1)
1.6 Reflection principles
11(1)
1.7 Analytic continuation
12(2)
1.8 The Stieltjes integral
14(1)
1.9 Hilbert spaces
15(2)
1.10 LP spaces
17(2)
Remarks and further reading
19(2)
2 Linear Fractional Transformations
21(12)
2.1 The Riemann sphere
21(4)
2.2 The cross-ratio and mapping properties of linear fractional transformations
25(2)
2.3 Upper half plane and unit disk
27(2)
Exercises
29(2)
Remarks and further reading
31(2)
3 Hyperbolic geometry
33(8)
3.1 Distance-preserving transformations and "lines"
33(1)
3.2 Construction of a distance function
34(3)
3.3 The triangle inequality
37(1)
3.4 Distance and area elements
38(1)
Exercises
39(1)
Remarks and further reading
40(1)
4 Harmonic functions
41(10)
4.1 Harmonic functions and holomorphic functions
41(1)
4.2 The mean value property, the maximum principle, and Poisson's formula
42(3)
4.3 The Schwarz reflection principle
45(1)
4.4 Application: approximation theorems
46(1)
Exercises
47(2)
Remarks and further reading
49(2)
5 Conformal maps and the Riemann mapping theorem
51(16)
5.1 Conformal maps
51(1)
5.2 The Riemann mapping theorem
52(2)
5.3 Proof of Lemma 5.2.2; the Ascoli--Arzela theorem
54(2)
5.4 Boundary behavior of conformal maps
56(1)
5.5 Mapping polygons: the Schwarz--Christoffel formula
57(2)
5.6 Triangles and rectangles
59(1)
5.7 Univalent functions
60(3)
Exercises
63(2)
Remarks and further reading
65(2)
6 The Schwarzian derivative
67(16)
6.1 The Schwarzian derivative as measure of curvature
67(2)
6.2 Some properties of the Schwarzian
69(1)
6.3 The Schwarzian and curves
70(1)
6.4 The Riemann mapping function and the Schwarzian
71(3)
6.5 Triangles and hypergeometric functions
74(3)
6.6 Regular polygons and hypergeometric functions
77(3)
Exercises
80(2)
Remarks and further reading
82(1)
7 Riemann surfaces and algebraic curves
83(22)
7.1 Analytic continuation
83(3)
7.2 The Riemann surface of a function
86(2)
7.3 Compact Riemann surfaces
88(1)
7.4 Algebraic curves: some algebra
89(4)
7.5 Algebraic curves: some analysis
93(2)
7.6 Examples: elliptic and hyperelliptic curves
95(2)
7.7 General compact Riemann surfaces
97(1)
7.8 Algebraic curves of higher genus
98(5)
Exercises
103(1)
Remarks and further reading
104(1)
8 Entire functions
105(16)
8.1 The Weierstrass product theorem
105(2)
8.2 Jensen's formula
107(2)
8.3 Functions of finite order
109(2)
8.4 Hadamard's factorization theorem
111(1)
8.5 Application to Riemann's xi function
112(3)
8.6 Application: an inhomogeneous vibrating string
115(3)
Exercises
118(1)
Remarks and further reading
119(2)
9 Value distribution theory
121(20)
9.1 The Nevanlinna characteristic and the first fundamental theorem
121(4)
9.2 The first fundamental theorem and a modified characteristic
125(3)
9.3 The second fundamental theorem
128(6)
9.4 Applications
134(3)
9.5 Further properties of meromorphic functions
137(1)
Exercises
138(2)
Remarks and further reading
140(1)
10 The gamma and beta functions
141(14)
10.1 Euler's product solution
141(3)
10.2 Euler's integral solution and the beta function
144(2)
10.3 Legendre's duplication formula
146(1)
10.4 The reflection formula and the product formula for sine
146(2)
10.5 Asymptotics of the gamma function
148(3)
Exercises
151(2)
Remarks and further reading
153(2)
11 The Riemann zeta function
155(12)
11.1 Properties of C
156(1)
11.2 The functional equation of the zeta function
157(3)
11.3 Zeros
160(1)
11.4 t(2m)
161(1)
11.5 The function ξ(s)
162(2)
Exercises
164(1)
Remarks and further reading
165(2)
12 L-functions and prunes
167(18)
12.1 Factorization and Dirichlet characters
168(1)
12.2 Characters of finite commutative groups
169(2)
12.3 Analysis of L-functions
171(2)
12.4 Proof of Dirichlet's Theorem
173(2)
12.5 Functional equations
175(5)
12.6 Other L-functions: algebraic and automorphic
180(2)
Exercises
182(1)
Remarks and further reading
183(2)
13 The Riemann hypothesis
185(20)
13.1 Primes and zeros of the zeta function
186(2)
13.2 Von Mangoldt's formula for Ψ
188(1)
13.3 The prime number theorem
189(3)
13.4 Density of the zeros
192(3)
13.5 The Riemann hypothesis and Gauss's approximation
195(2)
13.6 Riemann's 1859 paper
197(2)
13.7 Inverting the Mellin transform of Ψ
199(1)
Exercises
200(3)
Remarks and further reading
203(2)
14 Elliptic functions and theta functions
205(14)
14.1 Elliptic functions: generalities
205(4)
14.2 Theta functions
209(3)
14.3 Construction of elliptic functions
212(2)
14.4 Integrating elliptic functions
214(1)
Exercises
215(2)
Remarks and further reading
217(2)
15 Jacobi elliptic functions
219(10)
15.1 The pendulum equation
219(1)
15.2 Properties of the map F
220(2)
15.3 The Jacobi functions
222(3)
15.4 Elliptic curves: Jacobi parametrization
225(1)
Exercises
226(1)
Remarks and further reading
227(2)
16 Weierstrass elliptic functions
229(10)
16.1 The Weierstrass p function
229(3)
16.2 Integration of elliptic functions
232(2)
16.3 Elliptic curves: Weierstrass parametrization
234(1)
16.4 Addition on the curve
235(2)
Exercises
237(1)
Remarks and further reading
238(1)
17 Automorphic functions and Picard's theorem
239(16)
17.1 The elliptic modular function
239(1)
17.2 The modular group and the fundamental domain
240(3)
17.3 A closer look at λ Picard's theorem
243(4)
17.4 Automorphic functions; the J function
247(3)
Exercises
250(3)
Addendum: Moonshine
253(1)
Remarks and further reading
254(1)
18 Integral transforms
255(14)
18.1 Approximate identities and Schwartz functions
255(3)
18.2 The Cauchy Transform and the Hilbert transform
258(3)
18.3 The Fourier transform
261(1)
18.4 The Fourier transform for L1(R)
262(2)
18.5 The Fourier transform for L2(R)
264(1)
Exercises
265(3)
Remarks and further reading
268(1)
19 Theorems of Phragmen--Lindelof and Paley--Wiener
269(14)
19.1 Phragmen--Lindelof theorems
269(2)
19.2 Hardy's uncertainty principle
271(2)
19.3 The Paley--Wiener Theorem
273(5)
19.4 An application
278(2)
Exercises
280(1)
Remarks and further reading
281(2)
20 Theorems of Wiener and Levy; the Wiener--Hopf method
283(14)
20.1 The ring
283(3)
20.2 Convolution equations
286(4)
20.3 The case of real zeros of 1 -- k
290(2)
Exercises
292(2)
Remarks and further reading
294(3)
21 Tauberian theorems
297(18)
21.1 Hardy's theorem
298(1)
21.2 Abel, Tauber, Littlewood, and Hardy--Littlewood
299(2)
21.3 Karamata's tauberian theorem
301(3)
21.4 Wiener's tauberian theorem
304(5)
21.5 A theorem of Malliavin and applications
309(3)
Exercises
312(2)
Remarks and further reading
314(1)
22 Asymptotics and the method of steepest descent
315(16)
22.1 The method of steepest descent
315(2)
22.2 The Airy integral
317(2)
22.3 The partition function and the Hardy-Ramanujan formula
319(6)
22.4 Proof of the functional equation (22.3.6)
325(3)
Exercises
328(1)
Remarks and further reading
329(2)
23 Complex interpolation and the Riesz--Thorin theorem
331(10)
23.1 Interpolation: the complex method
331(3)
23.2 LP spaces
334(2)
23.3 Application: the Riesz-Thorin theorem
336(1)
23.4 Application to Fourier series
337(1)
Exercises
338(1)
Remarks and further reading
339(2)
References 341(8)
Index 349
Richard Beals is Professor Emeritus of Mathematics at Yale University. His research interests include ordinary and partial differential equations, operator theory, integrable systems, and transport theory.  He has authored many books, including Advanced Mathematical Analysis, published in 1973 as the twelfth volume in the series Graduate Texts in Mathematics.





Roderick S. C. Wong is Professor Emeritus of Mathematics at the City University of Hong Kong. His research interests include asymptotic analysis, perturbation methods, and special functions. He has been president of the Canadian Applied Mathematics Society and the Hong Kong Mathematical Society, and received numerous professional honors, including election to the European Academy of Sciences in 2007. He has written and edited a wide variety of books, with several notable works in the area of special functions.





This is the third book in the authors collaboration, after two previous volumeson special functions.