Mathematical Methods for Physics and Engineering : A Comprehensive Guide

12.9 EXERCISES

Deduce the value of the sumSof the series

1 −

1

33

+

1

53

−

1

73

+···.

12.15 Using the result of exercise 12.14, determine, as far as possible by inspection, the
forms of the functions of which the following are the Fourier series:

(a)

cosθ+

1

9

cos 3θ+

1

25

cos 5θ+···;

(b)

sinθ+

1

27

sin 3θ+

1

125

sin 5θ+···;

(c)

L^2

3

−

4 L^2

π^2

[

cos

πx

L

−

1

4

cos

2 πx

L

+

1

9

cos

3 πx

L

−···

]

.

(You may find it helpful to first setx= 0 in the quoted result and so obtain

values forSo=

∑

(2m+1)−^2 and other sums derivable from it.)
12.16 By finding a cosine Fourier series of period 2 for the functionf(t)thattakesthe
formf(t)=cosh(t−1) in the range 0≤t≤1, prove that
∑∞

n=1

1

n^2 π^2 +1

=

1

e^2 − 1

.

Deduce values for the sums

∑

(n^2 π^2 +1)−^1 over oddnand evennseparately.
12.17 Find the (real) Fourier series of period 2 forf(x)=coshxandg(x)=x^2 in the
range− 1 ≤x≤1. By integrating the series forf(x) twice, prove that
∑∞

n=1

(−1)n+1

n^2 π^2 (n^2 π^2 +1)

=

1

2

(

1

sinh 1

−

5

6

)

.

12.18 Express the functionf(x)=x^2 as a Fourier sine series in the range 0<x≤ 2
and show that it converges to zero atx=±2.
12.19 Demonstrate explicitly for the square-wave function discussed in section 12.2 that
Parseval’s theorem (12.13) is valid. You will need to use the relationship

∑∞

m=0

1

(2m+1)^2

=

π^2

8

.

Show that a filter that transmits frequencies only up to 8π/Twill still transmit
more than 90% of the power in such a square-wave voltage signal.
12.20 Show that the Fourier series for|sinθ|in the range−π≤θ≤πis given by

|sinθ|=

2

π

−

4

π

∑∞

m=1

cos 2mθ

4 m^2 − 1

.

By settingθ=0andθ=π/2, deduce values for

∑∞

m=1

1

4 m^2 − 1

and

∑∞

m=1

1

16 m^2 − 1

.