Stephen Karlson (shkarlson) wrote in 50bookchallenge,
Stephen Karlson
shkarlson
50bookchallenge

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THE JAGGED APPROXIMATED BY THE SMOOTH.

I had to finish Paul J. Nahin's An Imaginary Tale: The Story of i, reviewed here, to set up a review of his Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, a much delayed Book Review No. 9. Fabulous Formula is more technical than Imaginary Tale, and it might benefit by looking more like a textbook, with a few equation numbers so as to enable the reader to follow the frequent changes of variables Professor Nahin makes, generally with the end result being an equation of the form e = cos(θ) + i sin(θ), which can then be conveniently integrated, or split, or differentiated, depending on the result he's trying to derive. The formula turns up in a number of places, including Fourier series and Fourier integrals, and these transformations enable a practitioner to model phenomena such as pulses (the simple ones are a finite force over an interval of time, but more challenging ones including an instantaneous infinite force also lend themselves to modeling) and vibrating strings in a tractable way. The book is heavy going in places (keep a pencil with an eraser handy and take a lot of notes if you really wish to learn the methods) and more likely to appeal to the rocket scientist or electrical engineer than to the armchair mathematician or historian of mathematical thought (the biography of Euler in the final chapter notwithstanding). There are some results involving complex time that might repay careful study, if you're contemplating velocities that exceed the speed of light. If it's the history of mathematics you're after, Imaginary Tale is probably the better book for you.

(Cross-posted to Cold Spring Shops.)
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