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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