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Unformatted text preview: Fourier Series
When the French mathematician Joseph Fourier (1768–1830) was trying to solve a problem in heat conduction, he needed to express a function f as an inﬁnite series of sine and
fx 1 a0 a n cos n x bn sin n x n1 a0 a1 cos x a2 cos 2 x b1 sin x b2 sin 2 x a3 cos 3x
b3 sin 3x Earlier, Daniel Bernoulli and Leonard Euler had used such series while investigating problems concerning vibrating strings and astronomy.
The series in Equation 1 is called a trigonometric series or Fourier series and it turns
out that expressing a function as a Fourier series is sometimes more advantageous than
expanding it as a power series. In particular, astronomical phenomena are usually periodic,
as are heartbeats, tides, and vibrating strings, so it makes sense to express them in terms
of periodic functions.
We start by assuming that the trigonometric series converges and has a continuous function f x as its sum on the interval
, , that is, fx 2 a0 a n cos n x bn sin n x x n1 Our aim is to ﬁnd formulas for the coefﬁcients a n and bn in terms of f . Recall that for a
power series f x
cn x a n we found a formula for the coefﬁcients in terms of derivn
atives: cn f a n!. Here we use integrals.
If we integrate both sides of Equation 2 and assume that it’s permissible to integrate the
series term-by-term, we get y f x dx y y a0 dx a n cos n x bn sin n x d x n1 an y 2 a0 bn y sin n x d x cos n x d x n1 n1 But y cos n x d x 1
sin n x
n because n is an integer. Similarly, x y sin n x d x
f x dx 1
n sin n 0 0. So
2 a0 1 2 ■ FOURIER SERIES and solving for a0 gives
■ ■ Notice that a is the average value of f
over the interval 1
2 a0 3 y f x dx To determine an for n 1 we multiply both sides of Equation 2 by cos m x (where m is
an integer and m 1) and integrate term-by-term from
to : y y f x cos m x d x a0 a n cos n x bn sin n x cos m x d x n1 4 a0 y cos m x d x an y cos n x cos m x d x n1 bn y sin n x cos m x d x n1 We’ve seen that the ﬁrst integral is 0. With the help of Formulas 81, 80, and 64...
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