Mathematical Methods for Physics and Engineering : A Comprehensive Guide

12.9 EXERCISES

the sine and cosine form of the Fourier series, but the algebra is slightly more

complicated.

Parseval’s theorem is sometimes used to sum series. However, if one is presented

with a series to sum, it is not usually possible to decide which Fourier series

should be used to evaluate it. Rather, useful summations are nearly always found

serendipitously. The following example shows the evaluation of a sum by a

Fourier series method.

Using Parseval’s theorem and the Fourier series forf(x)=x^2 found in section 12.5,

calculate the sum

∑∞

r=1r

− (^4).
Firstly we find the average value of [f(x)]^2 over the interval− 2 <x≤2:
1
4

∫ 2

− 2

x^4 dx=

16

5

.

Now we evaluate the right-hand side of (12.13):

( 1

2 a^0

) 2

+^12

∑∞

1

a^2 r+^12

∑∞

1

b^2 n=

( 4

3

) 2

+^12

∑∞

r=1

162

π^4 r^4

.

Equating the two expression we find

∑∞

r=1

1

r^4

=

π^4

90

.

12.9 Exercises

12.1 Prove the orthogonality relations stated in section 12.1.
12.2 Derive the Fourier coefficientsbrin a similar manner to the derivation of thear
in section 12.2.
12.3 Which of the following functions ofxcould be represented by a Fourier series
over the range indicated?

(a) tanh−^1 (x), −∞<x<∞;

(b) tanx, −∞<x<∞;

(c)|sinx|−^1 /^2 , −∞<x<∞;

(d) cos−^1 (sin 2x), −∞<x<∞;

(e)xsin(1/x), −π−^1 <x≤π−^1 , cyclically repeated.

12.4 By moving the origin oftto the centre of an interval in whichf(t) = +1, i.e.
by changing to a new independent variablet′=t−^14 T, express the square-wave
function in the example in section 12.2 as a cosine series. Calculate the Fourier
coefficients involved (a) directly and (b) by changing the variable in result (12.8).
12.5 Find the Fourier series of the functionf(x)=xin the range−π<x≤π.Hence
show that

1 −

1

3

+

1

5

−

1

7

+···=

π

4

.

12.6 For the function

f(x)=1−x, 0 ≤x≤ 1 ,

find (a) the Fourier sine series and (b) the Fourier cosine series. Which would