Dr. Euler's Fabulous Formula by Nahin Paul J.;

Author:Nahin, Paul J.;
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2006-05-28T16:00:00+00:00

and explores what happens to them “mathematically” as T → ∞.

Perhaps “explore” is too gentle a word—what we’ll do is play with the two Fourier series equations in a pretty rough-and-ready way with little (if any) regard to justifying the manipulations. But—and this is important to understand—once we are done it won’t matter. Once we have the mathematical result that is our goal (i.e., the answer to “what happens as T → ∞?”) we can forget how we got it and simply treat it as a definition. The reason we can do this is because that result—called the Fourier integral—has deep physical significance, which is Nature’s way of telling us that, although we may have been a bit “casual” in getting to the result, the result is still a good result. Indeed, many books on the Fourier integral, written by mathematicians, take precisely this course of action.

Okay, let’s get started on seeing what happens as T → ∞. Notice first that the kω0 in the exponent of the ck integral changes by ω0 as k increases from one integer to the next. If we call this change Δω, then Δω = ω0. Now, since ω0 = 2π/T, as T → ∞ we see that ω0 → 0, that is, ω0 becomes arbitrarily small, and so we should write dω and not Δω as T → ∞, that is, as T → ∞ the change in the fundamental frequency becomes a differential change. Thus, for our first result, we have

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