1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.5 Operations on Fourier Series 87

Now, replacingbbyx,wehave

x( 2 π−x)

4

=

∑∞

n= 1

1

n^2

−

∑∞

n= 1

cos(nx)

n^2

, 0 ≤x≤ 2 π. (4)

Outside the indicated interval, the periodic extension of the function on the
left equals the series on the right.
It is worthwhile to mention that the series on the right of Eq. (4) is the
Fourier series of the function on the left. That is to say,

1

2 π

∫ 2 π

0

x( 2 π−x)

4 dx=

∑∞

n= 1

1

n^2 , (5)

1

π

∫ 2 π

0

x( 2 π−x)

4 cos(nx)dx=

− 1

n^2 , (6)

1

π

∫ 2 π

0

x( 2 π−x)

4

sin(nx)dx= 0. (7)

Equations (6) and (7) can be verified directly, of course, but Theorem 4, to-
gether with the orthogonality relations of Section 1, also guarantees them. In
addition, Eq. (5) gives us a way to evaluate the series on the right.

Although the uniqueness property stated in the following theorem is so very
natural that we tend to assume it is true without checking, it really is a conse-
quence of Theorem 4.

Theorem 5.If f(x)is periodic and sectionally continuous, its Fourier series is
unique.

That is to say, only one series can correspond tof(x).Weoftenmakeuseof
uniqueness in this way: If two Fourier series are equal (or correspond to the
same function), then the coefficients of like terms must match.
The last operation to be discussed is differentiation, one that plays a princi-
pal role in applications.

Theorem 6.If f(x)is periodic, continuous, and sectionally smooth, then the dif-
ferentiated Fourier series of f(x)converges to f′(x)at every point x where f′′(x)
exists:

f′(x)=

∑∞

n= 1

(

−nansin(nx)+nbncos(nx)

)

. (8)

The hypotheses onf(x)itself imply (see Section 4) that the Fourier series of
f(x)converges uniformly. Iff(x)(or its periodic extension) fails to be contin-