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Unformatted text preview: 4 sin 3t 3 4 4 sin 3t + sin 5t 3 5 are approximations to f (t) which improve as the number of terms increases. By the way, this Fourier series yields a curious formula for π . If we set x = π/2, f (x) = π , and we obtain φ5 (t) = 4 sin t + π = 4 sin(π/2) + 4 4 sin(3π/2) + sin(5π/2) + · · · 3 5 from which we conclude that π = 4(1 − 1111 + − + − · · · ). 3579 66 3 2 1 -4 -2 4 2 -1 -2 -3 Figure 3.2: A graph of the Fourier approximation φ13 (t). The overshooting near the points of discontinuity is known as the “Gibbs phenomenon.” The Fourier series on the right-hand side of (3.1) is often conveniently expressed in the Σ notation, f (t) = a0 + 2 ∞ ak cos( k=1 2πkt )+ T ∞ bk sin( k=1 2πkt ), T (3.9) just as we did for power series in Chapter 1. It is an interesting and diﬃcult problem in harmonic analysis to determine how “well-behaved” a periodic function f (t) must be in order to ensure that it can be expressed as a superposition of sines and cosines. An easily stated theorem, suﬃcient for...
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