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Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations

2018-07-19 Mathematics
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Release : 2018-07-19
Genre : Mathematics
Kind : eBook
Book Rating : 534/5 ( reviews)

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Book Synopsis Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations by : Niels Jacob

Download or read book Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations written by Niels Jacob. This book was released on 2018-07-19. Available in PDF, EPUB and Kindle. Book excerpt: In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures.Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard-Lindelöf and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm-Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows.Our introduction to the calculus of variations includes a discussion of the Euler-Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail.The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.

A Course in Analysis: Fourier analysis, ordinary differential equations, calculus of variations

2023 Calculus
Download A Course in Analysis: Fourier analysis, ordinary differential equations, calculus of variations PDF Online Free

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Release : 2023
Genre : Calculus
Kind : eBook
Book Rating : 147/5 ( reviews)

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Book Synopsis A Course in Analysis: Fourier analysis, ordinary differential equations, calculus of variations by : Niels Jacob

Download or read book A Course in Analysis: Fourier analysis, ordinary differential equations, calculus of variations written by Niels Jacob. This book was released on 2023. Available in PDF, EPUB and Kindle. Book excerpt:

A Course in Analysis

2016 Calculus
Download A Course in Analysis PDF Online Free

Author :
Release : 2016
Genre : Calculus
Kind : eBook
Book Rating : 511/5 ( reviews)

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Book Synopsis A Course in Analysis by : Niels Jacob

Download or read book A Course in Analysis written by Niels Jacob. This book was released on 2016. Available in PDF, EPUB and Kindle. Book excerpt: In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures. Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard-Lindelöf and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm-Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows. Our introduction to the calculus of variations includes a discussion of the Euler-Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail. The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.

Course In Analysis, A - Volume I: Introductory Calculus, Analysis Of Functions Of One Real Variable

2015-08-18 Mathematics
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Release : 2015-08-18
Genre : Mathematics
Kind : eBook
Book Rating : 106/5 ( reviews)

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Book Synopsis Course In Analysis, A - Volume I: Introductory Calculus, Analysis Of Functions Of One Real Variable by : Niels Jacob

Download or read book Course In Analysis, A - Volume I: Introductory Calculus, Analysis Of Functions Of One Real Variable written by Niels Jacob. This book was released on 2015-08-18. Available in PDF, EPUB and Kindle. Book excerpt: Part 1 begins with an overview of properties of the real numbers and starts to introduce the notions of set theory. The absolute value and in particular inequalities are considered in great detail before functions and their basic properties are handled. From this the authors move to differential and integral calculus. Many examples are discussed. Proofs not depending on a deeper understanding of the completeness of the real numbers are provided. As a typical calculus module, this part is thought as an interface from school to university analysis. Part 2 returns to the structure of the real numbers, most of all to the problem of their completeness which is discussed in great depth. Once the completeness of the real line is settled the authors revisit the main results of Part 1 and provide complete proofs. Moreover they develop differential and integral calculus on a rigorous basis much further by discussing uniform convergence and the interchanging of limits, infinite series (including Taylor series) and infinite products, improper integrals and the gamma function. In addition they discussed in more detail as usual monotone and convex functions. Finally, the authors supply a number of Appendices, among them Appendices on basic mathematical logic, more on set theory, the Peano axioms and mathematical induction, and on further discussions of the completeness of the real numbers. Remarkably, Volume I contains ca. 360 problems with complete, detailed solutions.

Course In Analysis, A - Vol. Ii: Differentiation And Integration Of Functions Of Several Variables, Vector Calculus

2016-06-29 Mathematics
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Release : 2016-06-29
Genre : Mathematics
Kind : eBook
Book Rating : 984/5 ( reviews)

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Book Synopsis Course In Analysis, A - Vol. Ii: Differentiation And Integration Of Functions Of Several Variables, Vector Calculus by : Niels Jacob

Download or read book Course In Analysis, A - Vol. Ii: Differentiation And Integration Of Functions Of Several Variables, Vector Calculus written by Niels Jacob. This book was released on 2016-06-29. Available in PDF, EPUB and Kindle. Book excerpt: 'The authors give many examples, illustrations and exercises to help students digest the theory and they employ use of clear and neat notation throughout. I really appreciate their selection of exercises, since many of the problems develop simple techniques to be used later in the book or make connections of analysis with other parts of mathematics. There are also solutions to all of the exercises in the back of the book. As in the first volume there are some real gems in volume II. A Course in Analysis seems to be full of these little gems where the authors use the material or ask the readers to use the material to obtain results or examples that the reader will certainly see again in another context later in their studies of mathematics. Generally, the quality of exposition in both of the first two volumes is very high. I recommend these books.' (See Full Review)MAA ReviewsThis is the second volume of 'A Course in Analysis' and it is devoted to the study of mappings between subsets of Euclidean spaces. The metric, hence the topological structure is discussed as well as the continuity of mappings. This is followed by introducing partial derivatives of real-valued functions and the differential of mappings. Many chapters deal with applications, in particular to geometry (parametric curves and surfaces, convexity), but topics such as extreme values and Lagrange multipliers, or curvilinear coordinates are considered too. On the more abstract side results such as the Stone-Weierstrass theorem or the Arzela-Ascoli theorem are proved in detail. The first part ends with a rigorous treatment of line integrals.The second part handles iterated and volume integrals for real-valued functions. Here we develop the Riemann (-Darboux-Jordan) theory. A whole chapter is devoted to boundaries and Jordan measurability of domains. We also handle in detail improper integrals and give some of their applications.The final part of this volume takes up a first discussion of vector calculus. Here we present a working mathematician's version of Green's, Gauss' and Stokes' theorem. Again some emphasis is given to applications, for example to the study of partial differential equations. At the same time we prepare the student to understand why these theorems and related objects such as surface integrals demand a much more advanced theory which we will develop in later volumes.This volume offers more than 260 problems solved in complete detail which should be of great benefit to every serious student.

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