FOURIER SERIES
be better for numerical evaluation? Relate your answer to the relevant periodic
continuations.
12.7 For the continued functions used in exercise 12.6 and the derived corresponding
series, consider (i) their derivatives and (ii) their integrals. Do they give meaningful
equations? You will probably find it helpful to sketch all the functions involved.
12.8 The functiony(x)=xsinxfor 0≤x≤πis to be represented by a Fourier series
of period 2πthat is either even or odd. By sketching the function and considering
its derivative, determine which series will have the more rapid convergence. Find
the full expression for the better of these two series, showing that the convergence
∼n−^3 and that alternate terms are missing.
12.9 Find the Fourier coefficients in the expansion off(x)=expxover the range
− 1 <x<1. What value will the expansion have whenx=2?
12.10 By integrating term by term the Fourier series found in the previous question
and using the Fourier series for∫ f(x)=xfound in section 12.6, show that
expxdx=expx+c. Why is it not possible to show thatd(expx)/dx=expx
by differentiating the Fourier series off(x)=expxin a similar manner?
12.11 Consider the functionf(x)=exp(−x^2 ) in the range 0≤x≤1. Show how it
should be continued to give as its Fourier series a series (the actual form is not
wanted) (a) with only cosine terms, (b) with only sine terms, (c) with period 1
and (d) with period 2.
Would there be any difference between the values of the last two series at (i)
x= 0, (ii)x=1?
12.12 Find, without calculation, which terms will be present in the Fourier series for
the periodic functionsf(t), of periodT, that are given in the range−T/2toT/ 2
by:
(a) f(t)=2for0≤|t|<T/ 4 ,f=1forT/ 4 ≤|t|<T/2;
(b)f(t)=exp[−(t−T/4)^2 ];
(c) f(t)=−1for−T/ 2 ≤t<− 3 T/8and3T/ 8 ≤t<T/ 2 ,f(t)=1for
−T/ 8 ≤t<T/8; the graph offis completed by two straight lines in the
remaining ranges so as to form a continuous function.
12.13 Consider the representation as a Fourier series of the displacement of a string
lying in the interval 0≤x≤Land fixed at its ends, when it is pulled aside byy 0
at the pointx=L/4. Sketch the continuations for the region outside the interval
that will
(a) produce a series of periodL,
(b) produce a series that is antisymmetric aboutx=0,and
(c) produce a series that will contain only cosine terms.
(d) What are (i) the periods of the series in (b) and (c) and (ii) the value of the
‘a 0 -term’ in (c)?
(e) Show that a typical term of the series obtained in (b) is
32 y 0
3 n^2 π^2
sinnπ
4sinnπx
L.
12.14 Show that the Fourier series for the functiony(x)=|x|in the range−π≤x<π
is
y(x)=π
2−
4
π∑∞
m=0cos(2m+1)x
(2m+1)^2.
By integrating this equation term by term from 0 tox, find the functiong(x)
whose Fourier series is
4
π∑∞
m=0sin(2m+1)x
(2m+1)^3