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Aimed at scientists and engineers, this book is an exciting intellectual journey through the mathematical worlds of Euclid, Newton, Maxwell, Einstein, and Schrodinger-Dirac.While similar books present the required mathematics in a piecemeal manner with tangential references to the relevant physics and engineering, this textbook serves the interdisciplinary needs of engineers, scientists and applied mathematicians by unifying the mathematics and physics into a single systematic body of knowledge but preserving the rigorous logical development of the mathematics.The authors take an unconventional approach by integrating the mathematics with its motivating physical phenomena and, conversely, by showing how the mathematical models predict new physical phenomena.
This Book Is Intended To Be A Simple And Easy Introduction To The Subject. It Is Meant As A Textbook For A Course In Complex Analysis At Postgraduate Level Of Indian Universities.Some Of The Welcome Features Of The Book Are: Proofs And Motivation For The Theory: Examples Are Provided To Illustrate The Concepts; Exercises Of Various Levels Of Difficulty Are Given At The End Of Every Chapter: Keeping In View The Applied Nature Of The Subject, Ordinary Linear Homogeneous Differential Equations Of The Second Order And Conformal Mapping And Its Applications Are Given More Attention Than Most Other Books: Uniform Approximation And Elliptic Functions Are Treated In Great Detail; There Is Also A Detailed Treatment Of Harmonic Functions, Weierstrass Approximation Theorem, Analytic Continuation, Riemann Mapping Theorem, Homological Version OfCauchys Theorem And Its Applications; Diagrams Are Provided Whenever Feasible To Help The Reader Develop Skill In Using Imagination To Visualise Abstract Ideas; Solutions To Some Selected Exercises Which Involve Lot Of New Ideas And Theoretical Considerations Have Been Provided At The End.
With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.