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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 righthand 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 “wellbehaved” 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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 Winter '14
 Equations

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