Mathematical Methods for Physics and Engineering : A Comprehensive Guide

12.5 NON-PERIODIC FUNCTIONS

− 220

L

x

f(x)=x^2

Figure 12.5 f(x)=x^2 ,0<x≤2, with the range extended to give periodicity.

the range, all the coefficientsbrwill be zero. Now we apply (12.5) and (12.6) withL=4
to determine the remaining coefficients:

ar=

2

4

∫ 2

− 2

x^2 cos

(

2 πrx

4

)

dx=

4

4

∫ 2

0

x^2 cos

(πrx

2

)

dx,

where the second equality holds because the function is even inx. Thus

ar=

[

2

πr

x^2 sin

(πrx

2

)]^2

0

−

4

πr

∫ 2

0

xsin

(πrx

2

)

dx

=

8

π^2 r^2

[

xcos

(πrx

2

)] 2

0

−

8

π^2 r^2

∫ 2

0

cos

(πrx

2

)

dx

=

16

π^2 r^2

cosπr

=

16

π^2 r^2

(−1)r.

Since this expression forarhasr^2 in its denominator, to evaluatea 0 we must return to the
original definition,

ar=

2

4

∫ 2

− 2

f(x)cos

(πrx

2

)

dx.

From this we obtain

a 0 =

2

4

∫ 2

− 2

x^2 dx=

4

4

∫ 2

0

x^2 dx=

8

3

.

The final expression forf(x)isthen

x^2 =

4

3

+16

∑∞

r=1

(−1)r

π^2 r^2

cos

(πrx

2

)

for 0<x≤ 2 .

We note that in the above example we could have extended the range so as

to make the function odd. In other words we could have setf(x)=−f(−x)and

then madef(x) periodic in such a way thatf(x+4) =f(x). In this case the

resulting Fourier series would be a series of just sine terms. However, although

this will faithfully represent the function inside the required range, it does not