The Functions of Mathematical Physics (Dover Books on Physics) by Harry Hochstadt (PDF)

Description

A modern classic, this clearly written, incisive textbook provides a comprehensive, detailed survey of the functions of mathematical physics, a field of study straddling the somewhat artificial boundary between pure and applied mathematics.In the 18th and 19th centuries, the theorists who devoted themselves to this field — pioneers such as Gauss, Euler, Fourier, Legendre, and Bessel — were searching for mathematical solutions to physical problems. Today, although most of the functions have practical applications, in areas ranging from the quantum-theoretical model of the atom to the vibrating membrane, some, such as those related to the theory of discontinuous groups, still remain of purely mathematical interest.Chapters One and Two examine orthogonal polynomials, with sections on such topics as the recurrence formula, the Christoffel-Darboux formula, the Weierstrass approximation theorem, and the application of Hermite polynomials to quantum mechanics.Chapter Three is devoted to the principal properties of the gamma function, including asymptotic expansions and Mellin-Barnes integrals. Chapter Four covers hypergeometric functions, including a review of linear differential equations with regular singular points, and a general method for finding integral representations.Chapters Five and Six are concerned with the Legendre functions and their use in the solutions of Laplace’s equation in spherical coordinates, as well as problems in an n-dimension setting. Chapter Seven deals with confluent hypergeometric functions, and Chapter Eight examines, at length, the most important of these — the Bessel functions. Chapter Nine covers Hill’s equations, including the expansion theorems.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book certainly provided a good overview of the functions of mathematical physics, and it used some unconventional arguments that were enjoyable to read. On the other hand, I found the exposition and notation just slightly lacking–in some sections, a few extra words or notational clarification would have clarified things immensely. One example is that in the early part of the book, Hochstadt defines Legendre polynomials over an interval, and then uses identical notation in his characterization of the Lagrange interpolation functions, which are defined for a particular set of zeros. I found this apparent overlap of notation confusing. As a consequence, I’ve had to annotate my own copy of this book pretty heavily.Otherwise, the book is very practical, and for me, it’s true value was in providing some of the context and unifying theory for special functions that many other books on the subject lack. For a more systematic approach to special functions, I have found Lebedev’s book to be a useful complement to this one. Truesdell also wrote a nice monograph “An Essay Toward a Unified Theory of Special Functions” that characterizes most special functions by a certain functional form.

⭐This is a good concise book concerning with those functions relevant to physical applications. However, there is a mistkake at page 61 about the defintion of Gamma fuctions. In fact the second defintion is also due to Euler, and not Gauss. Gauss’s one is the Factorial fuction. I hope in the future if the book is revised, this mistake can be corrected.Another mistake (probably printing mistake ) occurs in page 62,line six. uk should be equal to I/k – ln(k+1/k ).

⭐A must read for those studying physics or math. The section on the spherical harmonic functions is the shining star of this work.

⭐This is a very nice introduction to special functions, which historically have seen important applications in Physics and Applied Mathematics.Functions and Equations studied include: Orthogonal Polynomials, Gamma Function, Hypergeometric Functions, Legendre Polynomials, Bessel Functions, Hill Equation.I would recommend this book to 2 kinds of people:a) Students of Physics and Applied Math;b) Students of PURE mathematics, who having completed studying a first rigorous course in Real Anaysis, PLUS a course in Complex Analysis [Holomorphic Functions, Conformal Mapping, Analytic Continuations, etc] , want to apply whatever analytical agility they will have acquired. This sort of person will find herself applying a lot of the analysis she has learned.

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