Transform Methods

about mathematical physics methods and special functions

First, in the era before Mathematica, physicists and engineers often had to consult handbook tables to derive formulas; even now that Mathematica exists, having such references on the desk is still useful, and moreover Mathematica has its own limitations.

Second, no matter how elegant the theory, in practice one must use computers to compute; therefore one cannot be satisfied with merely deriving some closed-form formulas. To truly solve problems, numerical methods are indispensable, and error estimates must be provided.

Fourier Transform Methods in Finance https://books.ms/main/0BA6516E92D3CD7727F2A60A8185AB61 and Transform Methods for Solving Partial Differential Equations https://books.ms/main/903A94E3799B40E020664A18FE2A060E provide concrete examples for the above views. The former is applied directly to financial modeling, while the author of the latter, Dr. Dean Duffy, received his doctorate from MIT and long served the U.S. military (the United States Naval Academy, the United States Military Academy, and the U.S. Air Force), and is now an engineer at NASA. His book on integral transforms and another book on Green's functions (Green's functions with Applications https://books.ms/main/2A8B921E4A7E76123A739E384F26722A) are extremely practical and targeted, solving real problems he and his colleagues encountered at work. Another characteristic of his books is that they provide "one-stop" solutions: not only theoretical formula derivations, but more importantly numerical-computation solutions.