~

EXERCISES 765

a 00

f(x) = ---9.+ I elncos(nx - en),2 n=1

Show that this is equivalent to Eq. 14.1with

an = elnCOSen, el;= a; + b;,

bn= elnsin en, tanen=b,jan.

Note. The coefficients el;as a function of n define what is called the power spec-trum. The importance of el; lies in its invariance under a shift in the phase en,

14.1.3 A function f(x) is expanded in an exponential Fourier series00

f(x) = I cneinx.

If f(x) is real, f(x) = f*(x), what restriction is imposed on the coefficients cn?

14.1.4 Assuming that S~nf(x) dx and S~"[f(X)J2 dx are finite, show that

lim am= O,m-oo l~ bm = O.

Hirtt.Integrate [f(x) - sn(x)J2,wheresn(x)is the nth partial sumand useBessel'sinequality, Section 9.4. For our finite interval the assumption that f(x) is squareintegrable (S~"If(x)!l dx is finite) implies tha0" I (Ix)!dx is also finite. Theconverse does not boldo

f(x) J) (Jot.I.(:t..\t~.,.6

'31 14. /. f

-rr

00

f(x) =¡ ~ sin nxn=1

xrr

rr- 2

FIG. 14.2

14.1.5 Apply the summation technique of this section to show that

I sinnx ={

1(11:- x), O < x:::; 11:

n=1 n -1(11: + x), -11: :::;X < O

(Fig. 14.2).

14.1.6 Sum the trigonometric series

766 FOURIER SERIES

00

I (_1)n+1 sinnxn~l n

and show that it equals x/2.

14.1 .7 Sum the trigonometric series

I sin(2n + l)xn~O 2n + 1 '

and show that it equals

{

n/4,

-n/4,

O<x<n

- n < x < O.

14.1.8 Calculate the sum ofthe finite Fourier sine series for the sawtooth wave, f(x) = x,(-n, n), Eq. 14.15. Use 4-, 6-, 8-, and 10-term series and x/n = 0.00(0.02)1.00.If a plotting routine is available, plot your results and compare with Fig. 14.1.

14.2 ADV ANT AGES. USES OF FOURIER SERIES

Discontinuous Function

One of the advantages of a Fourier representation ayer some other represen-tation, such as a Taylor series, is that it mar represent a discontinuous function.An example is the sawtooth wave in the preceding section. Other examples areconsidered in Section 14.3 and in the exercises.

Periodic Functions

Related to this advantage is the usefulness of a Fourier series in representinga periodic function. Uf(x) has a period of 2n, perhaps it is only natural that weexpand it in a series of functions with period 2n, 2n/2, 2n/3, . . . . This guaranteesthat if our periodic f(x) is represented ayer one interval [O,2n] or [ - n, n] therepresentation holds for all finite x.

At this point we mar conveniently consider the properties ofsymmetry. Usingthe interval [ - n,n], sinx is odd and cosx is an even function of x. Hence byEqs. 14.11 and 14.12,1 iff(x) is odd, all Gn= O and iff(x) is even all bn = O.Inother words,

G 00

f(x) = --2 + I Gncos nx,2 n~1

f(x) even, (14.21)

00

f(x) = I bnsin nx,n=1

f(x) odd. (14.22)

Frequently these properties are helpful in expanding a given function.We have noted that the Fourier series is periodic. This is important in con-

sidering whether Eq. 14.1holds outside the initial interval. Suppose we are givenonly that

lWith the range ofintegration -n:$; x:$; n.

--

EXERCISES 769

replacing x in Eq. 14.1 with nx/L and t in Eqs. 14.11 and 14.12 with nt/L. (Forconvenience the interval in Eqs. 14.11 and 14.1t' i!':!':hiftedto -n ::;;t::;;n.) Thechoice of the syrnrnetric interval (- L, L) is notperiod of 2L, any interval (xo, Xo + 2L) will dvenience or literally personal preference.

A) fJ.\,(.~

EXERCISES

14.2.1 The boundary conditions(suchas tf¡(0)= tf¡(l)= O)may suggestsolutionsof theforro sin(nnxjl) and eliminate the corresponding cosines.(a) Verify that the boundary conditions used in the Sturm-Liouville theory

are satisfied for the interval (O,1).Note that this is only halfthe usual Fourierinterval.

(b) Show that the set of functions <Pn(x)= sin(nnxjl), n = 1, 2, 3, ... satisfiesan orthogonality relation

14.2.2 (a)

11 1

<Pm(x)<Pn(x)dx = -13mn,o 2

Expand f(x) = x in the interval (0,2L). Sketch the series you have found(right-hand side of Ans.) ayer (- 2L, 2L).

2L 00 1 .

(nnx

)ANS. x=L-- I -SIn - .n n=1n L

n> O.

(b) Expand f(x) = x as a sine series in the half interval (O,L). Sketch the seriesyou have found (right-hand side of Ans.) ayer (- 2L, 2L).

ANS. x=2L I(-I)"+1sin (nnx

).n Fl L

14.2.3 In some problems it is convenient to approximate sin nx ayer the interval [0,1]by a parabola ax(1 - x), where a is a constant. To get a feeling for the accuracyofthis approximation, expand 4x(1 - x) in a Fourier sine series:

{4X(1 - x), 8 < x < 1

}

00

f(x)= 4x(l+x), -1~x~O =n~lbnsinnnx

32 1ANS. bn=3"3'n n

bn = O,

n odd

n even(Fig. 14.4).

f(x)

x

FIG. 14.4

." ..",-

774 FOURIER SERIES

EXERCISES

'er series representation of

z;) \'\. ~i z..

G) \'-'1-)' c¡-r) ¡'-\' 1- \'1

14.3.2

14.3.3

14.3.4

14.3.5

{

O,

f(t) = sin wt,

- n S wt S O,

OS wt S n.

of a simple half-wave rectifier. It is also an approximational effect that produces "tides" in the atmosphere.

1 1. 2 00 cos nwtANS. f(t) = - + -smwt - - ¿ z--.

n 2 n n=2,4,6.. n - 1even

A sawtooth wave is given by

f(x) = x, -n < x < n.Show that

00 (-1)"+1f(x)=2 ¿ -sinnx.

n~1 n

A different sawtooth wave is described by

f(x) ={ -!(n + x),

+z(n - x),

Show that f(x) = ¿;;'~1(sinnxjn).

A triangular wave (Fig. 14.7) is represented by

-nsx<O

O < x S n.

f(x) ={

X,

-x,

O<x<n

-n < x < O.

Represent f(x) by a Fourier series.

ANS. f(x) =~ - ~ ¿ cosnx2 n n=1,3 5,.. n2 .

odd

f(x)

--x-4." -3." -2." 3."2." 4."-." ."

FIG.14.7 Triangular wave

Expand

{1'

f(x) = O,x2 < x6

X2 > x6

in the interval [ - n, n].

---

14.3.6

14.3.7

14.3.8

14.3.9

14.3.10

EXERCISES 775

nu FIG. 14.8

Note. This variable width square wave is of some importance in electronicmusic.

A metal cylindrical tube of radius a is split lengthwise into two nontouchinghalyes. The top half is maintained at a potential + V, the bottom half at apotential - V (Fig. 14.8).Separate the variables in Laplace's equation and salvefor the electrostatic potential for r :::;a. Observe the resemblance between yoursolution for r = a and the Fourier series for a square wave.

A metal cylinder is placed in a (previously) uniform electric field, Eo, the axisof the cylinder perpendicular to that of the original field.(a) Find the perturbed electrostatic potential.(b) Find the induced surface charge on the cylinder as a function of angular

position.

Transform the Fourier expansion of a square wave, Eq. 14.3.6, into a powerseries. Show that the coefficients of Xl forro a divergent series. Repeat for thecoefficients of X3.

A power series cannot handle a discontinuity. These infinite coefficients arethe result of attempting to beat this basic limitation on power series.

Show that the Fourier expansion of cos ax is

cosax = 2asinan{~ - ~osx + cos2x - .. .

}n 2a2 a2 - 12 a2 - 22 '

an = (-1)" 2a sin an .n(a2 - n2)

(b) From the preceding result show that

(a)

00

ancotan = 1 - 2 I ((2p)a2P.p=l

This pro vides an alternate derivation of the relation between the Riemannzeta function and the Bernoulli numbers, Eq. 5.151.

Derive the Fourier series expansion of the Dirac delta function c5(x)in theinterval -n < x < n.(a) What significance can be attached to the constant term?(b) In what regían is this representation valid?(c) With the identity

.....

I

¡

deriveEXERCISES 777

and show that

8 00

f(x)=- ¿ ~10 n~1,3,5,.. n3

odd

00 1 1 1 103

¿ (-1)(n-1)/2n-3 = 1 - 33 + 53 - 73 + . . . = 32 = p(3).n~1,3,5,..

odd

(e)

Using the Fourier series for a square wave, show that

f (_lyn-1)/2n-1 = 1 - ~+ ~- ~+ . . . = !:= {3(1).n~l 3 5 3 5 7 4o'de! '"

14,3.14 (a) Find the Fourier series representation of

This is Leibnitz's formula for 10,obtained by a different technique in Exercise5.7.6.

Note. The 1'/(2),1'/(4),),(2), {3(1),and {3(3)functions are defined by the indi-cated series. General definitions appear in Section 5.9.

(b) From your Fourier expansion show that

102 1 1-=1+-+_+....8 32 52

f(x) ={

O,

x,-n<x:$O

0:$ x < n.

14.3.15

Let fez) = In(l + z) = ¿;:;"1(-1t+1zn/n. (This series converges to In(l + z) for)zl :$ 1, except at the point z = -1.)(a) From the imaginary parís show that

I(2

8

)~

( 1)n+1 cosn8n cos- = L.. - -2 n~l n '

(b) Using a change of variable, transform part (a) into

( . q»00 cos nq>

-In 2 sm- = ¿ ~, O< q> < 210.2 n~l n

- 10 < 8 < n.

14.3.16 A symmetric triangular pulse of adjustable height and width is described by

{a(l - x/b), 0:$ Ix! :$ bf(x) =O, b:$ Ix) :$ n.

(a) Show that the Fourier coefficients are

ao = ab10'

2aban = -(1 - cosnb)/(nbf.10

Sum the finite Fourier series through n = 10 and through n = 100 forx/n = 0(1/9)1. Take a = 1 and b = 10/2.

(b) Cal! a Fourier analysis subroutine (if available) to calculate the Fouriercoefficients off(x), ao through ala'

14.3.17 (a) Using a Fourier analysis subroutine, calculate the Fourier cosine coeffi-cients ao through ala of

f(x) = [1 - (x/n)2J1/2, [-10,10].

EXERCISES 779

Ix f(x) dx - !aoxXo 1)

,~ '-( ,

will still be a Fourier series.

Differentiation I

The situation regarding differentiation is quite difLtion. Here the word is caution. Consider the series for

f(x) = x, - n < x < n. (14.54)

We readily find (compare Exercise 14.3.2)that the Fourier series is00

x = 2 L ( -lt+1 sin nxn=l n '

- n < x < n. (14.55)

Differentiating term by term, we obtain00

1 = 2 L (-lt+1cosnx,n=l

(14.56)

which is not convergent! Warning. Check your derivative.For a triangular wave (Exercise 14.3.4), in which the convergence is more

rapid (and uniform),

f(x) =?E - ~ ~ cosnx2 n L... z- .n=l,odd n(14.57)

Differentiating term by term

f'(x) = ~ f Slll nxnn=l,odd n

(14.58)

which is the Fourier expansion of a square wave

{

1,

f'(x) = -1,

O<x<n,

- n < x < O.(14.59)

Inspection of Fig. 14.7verifies that this is indeed the derivative of our triangularwave.

As the inverse of integration, the operation of differentiation has placed anadditional factor n in the numerator of each termo This reduces the rate ofconvergence and may, as in the first case mentioned, render the differentiatedseries divergent.

In general, term-by-term differentiation is permissible under the same condi-tions listed for uniform convergence.

EXERCISES

14.4.1 Show that integration of the Fourier expansion of f(x) = x, -n < x < n, leadsto

780

14.4.2

14.4.3

II 1 I 1",I ,

FOURIER SERIES

z 00

~= ¿ (_1)n+1n-Z12 n-1

=1-t+~-l6+ "'.

Parseval's identity.(a) Assuming that the Fourier expansion off(x) is uniformly convergent, show

that

1

fn Z 00

- [J(x)]Z dx = ao + ¿ (a; + b;).n -n 2 n-1

This is Parseval's identity. It is actually a special case of the completenessrelation, Eq. 9.72.

(b) Given

nZ 00 (-l)ncosnxXZ = - + 4 ¿ Z '

3 n=l n

apply Parseval's identity to obtain '(4) in closed formo(c) The condition of uniform convergence is not necessary.

applying the Parseval identity to the square wave-n<x<O

O<x<n

-n ~ x ~ n,

Show this by

{

-1,

f(x) = 1,

= ~ f sin(2n - l)x.n n=l 2n - 1

Show that integrating the Fourier expansion of the Dirac delta function(Exercise 14.3.10) leads to the Fourier representation of the square wave, Eq.14.3.6, with h = 1.Note. Integratingthe constant term (1/2n)leads to a term x/2n. What are yougoing to do with this?

14.4.3A Integrate the Fourier expansion of the unit step function

14.4.4

14.4.5

f(x) ={

O,

x,

-n < x < O

O < x < n.

Show that your integrated series agrees with Exercise 14.3.14.

In the interval (- n, n),

cin(x)= n,1

for Ixl <-,2n

1for Ixl > 2nO,

(Fig. 14.9).(a) Expands cin(x)as a Fourier cosine series.(b) Show that your Fourier series agrees with a Fourier expansion of ci(x)in

the limit as n -+ oo.

Confirm the delta function nature of your Fourier series of Exercise 14.4.4byshowing that for any f(x) that is finite in the interval [ - n, n] and continuousat x = O,

fn f(x) [Fourier expansion of Cioo(x)]dx = f(O).

---"""118