Bateman H. Partial Differential Equations of Mathematical Physics

New York: Dover, 1944. — 537 p.Harry Bateman (1882-1946) was an esteemed mathematician particularly known for his work on special functions and partial differential equations. This book, first published in 1932, has been reprinted many times and is a classic example of Bateman's work. Partial Differential Equations of Mathematical Physics was developed chiefly with the aim of obtaining exact analytical expressions for the solution of the boundary problems of mathematical physics.The book is really concerned with second-order partial differetial equation (PDE) boundary value problems (BVP), since at that time (1932) these were often used to model physical processes (e.g. the heat equation, the wave equation). This is NOT an elementary textbook: it assumes the reader is already familiar with the basics of 2nd-order PDEs, and there are very few mathematical proofs. The level is quite advanced (for 1932).The format of this book reminds me of Ince, Ordinary Differential Equations, 1923. But Bateman is more of a compendium of results, and in this sense it is like Abramowitz and Stegun, Handbook of Mathematical Functions, 1955. On the positive side, it contains a huge amount of material. It is organized around the type of trick used to solve the PDE BVP. Several of the chapters are devoted to unusual (but very useful) coordinate systems. If there is any common tool, it is the Green's function, which appears throughout the book.There are some problems with obsolete terminology. These are aggrivated by Bateman's lack of RIGOROUS definitions. For example, he uses C for the class of functions with continuous second partial derivatives. A modern work would use a superscript of 2 on the C, and indicate the domain of applicability. Indeed, except in rare instances, Bateman does not give the domain of applicability. Also, the term "eit" is used for eigenvalue.Over the years I have worked with second-order PDE BVPs, using hyperbolic PDEs, parabolic PDEs and elliptic PDEs. But I cannot say that this book ever helped with a solution. For 2nd order PDEs have moved on considerably since 1932, and the methods I use differ (e.g. the now-3-decades-old split-step FFT for a parabolic equation). I like this book, but it is now quite dated. Read this book for a good background on how PDE BVPs can be solved. It is good reading. But also read a more modern account before you actually go off to solve a real-world problem.