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Unformatted text preview: a jump discontinuity, the onesided limits exist and we use the notation
fa lim f x fa xla lim f x xla 8 Fourier Convergence Theorem If f is a periodic function with period 2 and f and
f are piecewise continuous on
, , then the Fourier series (7) is convergent.
The sum of the Fourier series is equal to f x at all numbers x where f is continuous. At the numbers x where f is discontinuous, the sum of the Fourier series is
the average of the right and left limits, that is
1
2 fx fx If we apply the Fourier Convergence Theorem to the squarewave function f in
Example 1, we get what we guessed from the graphs. Observe that
f0 lim f x 1 xl0 and f0 lim f x 0 xl0 and similarly for the other points at which f is discontinuous. The average of these left and
right limits is 1 , so for any integer n the Fourier Convergence Theorem says that
2
1
2 2
k1 2k 1 sin 2k (Of course, this equation is obvious for x 1x
n .) fx
1
2 if n
if x n
n 6 ■ FOURIER SERIES Functions with Period 2L
If a function f has period other than 2 , we can ﬁnd its Fourier series by making a change
f x for all x. If we let
of variable. Suppose f x has period 2 L, that is f x 2 L
t
x L and
tt fx f Lt then, as you can verify, t has period 2 and x
series of t is
a0 L corresponds to t a n cos nt . The Fourier bn sin nt n1 where
1
2 a0
1 an y y t t dt t t cos nt dt If we now use the Substitution Rule with x
we have the following
9 Lt nx
L a n cos
n1 y t t sin nt dt , then t If f is a piecewise continuous function on
a0 1 bn x L, dt L d x, and L, L , its Fourier series is
bn sin nx
L where
Notice that when L
these formulas
are the same as those in (7).
■■ 1
2L a0
and, for n
an 1
L y L f x dx L 1, y L
L f x cos nx
L dx 1
L bn y L
L f x sin nx
L dx Of course, the Fourier Convergence Theorem (8) is also valid for functions with period
2 L.
EXAMPLE 2 Find the Fourier series of the triangular wave function deﬁned by f x
x
for 1 x 1 and f x 2
f x for all x. (The graph of f is shown in Figure 3.)
For which values of x is f x equa...
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This note was uploaded on 12/05/2011 for the course MATH 41 taught by Professor Bray,c during the Spring '08 term at Duke.
 Spring '08
 Bray,C
 Math, Fourier Series

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