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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.

Complex Analysis and Differential Equations

2012-04-23 Mathematics
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Release : 2012-04-23
Genre : Mathematics
Kind : eBook
Book Rating : 087/5 ( reviews)

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Book Synopsis Complex Analysis and Differential Equations by : Luis Barreira

Download or read book Complex Analysis and Differential Equations written by Luis Barreira. This book was released on 2012-04-23. Available in PDF, EPUB and Kindle. Book excerpt: This text provides an accessible, self-contained and rigorous introduction to complex analysis and differential equations. Topics covered include holomorphic functions, Fourier series, ordinary and partial differential equations. The text is divided into two parts: part one focuses on complex analysis and part two on differential equations. Each part can be read independently, so in essence this text offers two books in one. In the second part of the book, some emphasis is given to the application of complex analysis to differential equations. Half of the book consists of approximately 200 worked out problems, carefully prepared for each part of theory, plus 200 exercises of variable levels of difficulty. Tailored to any course giving the first introduction to complex analysis or differential equations, this text assumes only a basic knowledge of linear algebra and differential and integral calculus. Moreover, the large number of examples, worked out problems and exercises makes this the ideal book for independent study.

Real Analysis and Applications

2021-10-25 Mathematics
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Release : 2021-10-25
Genre : Mathematics
Kind : eBook
Book Rating : 019/5 ( reviews)

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Book Synopsis Real Analysis and Applications by : Frank Morgan

Download or read book Real Analysis and Applications written by Frank Morgan. This book was released on 2021-10-25. Available in PDF, EPUB and Kindle. Book excerpt: Real Analysis and Applications starts with a streamlined, but complete approach to real analysis. It finishes with a wide variety of applications in Fourier series and the calculus of variations, including minimal surfaces, physics, economics, Riemannian geometry, and general relativity. The basic theory includes all the standard topics: limits of sequences, topology, compactness, the Cantor set and fractals, calculus with the Riemann integral, a chapter on the Lebesgue theory, sequences of functions, infinite series, and the exponential and Gamma functions. The applications conclude with a computation of the relativistic precession of Mercury's orbit, which Einstein called "convincing proof of the correctness of the theory [of General Relativity]." The text not only provides clear, logical proofs, but also shows the student how to come up with them. The excellent exercises come with select solutions in the back. Here is a text which makes it possible to do the full theory and significant applications in one semester. Frank Morgan is the author of six books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this applied version of his Real Analysis text, Morgan brings his famous direct style to the growing numbers of potential mathematics majors who want to see applications right along with the theory.

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