Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers
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6.14 Exercises 221
3. Obtain the Fourier cosine series representation for the following func-
tions:
(a) f (x)=π + x 0 <x<π,
(b) f (x)=x 0 <x<π,
(c) f (x)=x
2
0 <x<π,
(d) f (x)=sin3x 0 <x<π,
(e) f (x)=e
x
0 <x<π,
(f) f (x) = cosh x 0 <x<π.
4. Expand the following functions in a Fourier series:
(a) f (x)=x
2
+ x −1 <x<1,
(b) f (x)=
⎧
⎨
⎩
1
0
0 <x<3
3 <x<6,
(c) f (x)=sin(πx/l)0<x<l,
(d) f (x)=x
3
−2 <x<2,
(e) f (x)=e
−x
0 <x<1,
(f) f (x)=sinhx −1 <x<1.
5. Expand the following functions in a complex Fourier series:
(a) f (x)=e
2x
−π<x<π,
(b) f (x) = cosh x −π<x<π,
(c) f (x)=
⎧
⎨
⎩
1
cos x
−π<x<0
0 <x<π,
(d) f (x)=x −1 <x<1,
(e) f (x)=x
2
−π<x<π,
(f) f (x)=sinh(πx/2) −2 <x<2.
222 6 Fourier Series and Integrals with Applications
6. (a) Find the Fourier series expansion of the function
f (x)=
⎧
⎨
⎩
0, −π<x<0
x/2, 0 <x<π.
(b) With the use of the Fourier series of f (x) in 6(a), show that
π
2
8
=1+
1
3
2
+
1
5
2
+
1
7
2
+ ....
7. (a) Determine the Fourier series of the function
f (x)=x
2
, −l<x<l.
(b) With the use of the Fourier series of f (x) in 7(a), show that
π
2
12
=1−
1
2
2
+
1
3
2
−
1
4
2
+ ....
8. Determine the Fourier series expansion of each of the following functions
by performing th e differentiation of the appropriate Fourier series:
(a) sin
2
x 0 <x<π,
(b) cos
2
x 0 <x<π,
(c) sin x cos x 0 <x<π,
(d) cos x + cos 2x 0 <x<π,
(e) cos x + cos 2x 0 <x<π.
9. Find the functions represented by the new series which are obtained by
termwise integration of the following series from 0 to x:
(a)
∞
k=1
(−1)
k+1
k
sin kx = x/2 −π<x<π,
(b)
3
2
+
1
π
∞
k=1
1−(−1)
k
k
sin kx =
⎧
⎨
⎩
1
2
−π<x<0
0 <x<π,
(c)
∞
k=1
(−1)
k+1
cos kx
k
=ln
2cos
x
2
−π<x<π,
6.14 Exercises 223
(d)
∞
k=1
sin(2k+1)x
(2k+1)
3
=
π
2
x−πx
2
8
0 <x<2π,
(e)
4
π
∞
k=1
sin(2k−1)x
(2k−1)
=
⎧
⎨
⎩
−1
1
−π<x<0
0 <x<π.
10. Determine the double Fourier series of the following functions:
(a) f (x, y)=1 0<x<π 0 <y<π,
(b) f (x, y)=xy
2
0 <x<π 0 <y<π,
(c) f (x, y)=x
2
y
2
0 <x<π 0 <y<π,
(d) f (x, y)=x
2
+ y −π<x<π −π<y<π,
(e) f (x, y)=x sin y −π<x<π −π<y<π,
(f) f (x, y)=e
x+y
−π<x<π −π<y<π,
(g) f (x, y)=xy 0 <x<10<y<2,
(h) f (x, y)=1 0<x<a 0 <y<b,
(i) f (x, y)=x cos y −1 <x<1 −2 <y<2,
(j) f (x, y)=xy
2
−π<x<π −π<y<π,
(k) f (x, y)=x
2
y
2
−π<x<π −π<y<π.
11. Deduce the general double Fourier series expansion f ormula for the func-
tion f (x, y) in the rectangle −a<x<a, −b<y<b.
12. Prove the Weierstrass Approximation Theorem:Iff is a continuous
function on the interval −π ≤ x ≤ π and if f (−π)=f (π), then, for
any ε>0, there exists a trigonometric polynomial
T (x)=
a
0
2
+
n
k=1
(a
k
cos kx + b
k
sin kx)
such that
|f (x) − T (x)| <ε
for all x in [−π, π].
224 6 Fourier Series and Integrals with Applications
13. Use the Fourier cosine or sine integral formula to show that
(a) e
−αx
=
2
π
∞
0
α
α
2
+β
2
cos βxdβ, x ≥ 0, α>0,
(b) e
−αx
=
2
π
∞
0
β
α
2
+β
2
sin βx dβ, x>0, α>0.
14. Show that the Fourier integral representation of the function
f (x)=
⎧
⎨
⎩
x
2
, 0 <x<a
0,x>a
is
f (x)=
2
π
∞
0
a
2
−
2
k
2
sin ak +
2a
k
cos ak
cos kx
k
dk.
15. Apply the Parseval relation (6.5.10) to Example 6.7.3 or Example 6.7.4
to show that
(a)
∞
n=1
1
(2n − 1)
4
=
π
4
96
and (b)
∞
n=1
1
n
4
=
π
4
90
.
16. (a) Obtain the Fourier series for the 2π-periodic odd function f (x)=
x (π − x)on[0,π].
(b) Use the Parseval relation (6.5.10) to show that
∞
n=1
1
(2n − 1)
6
=
π
6
960
and
∞
n=1
1
n
6
=
π
6
945
.
17. If the 2π-periodic even function is given by f (x)=|x| for −π ≤ x ≤ π,
show that
f (x)=
π
2
−
4
π
∞
n=1
cos (2n − 1) x
(2n − 1)
2
.
18. Consider the sawtooth function defined by f (x)=π − x,0<x<2π,
and f (x +2nπ)=f (x)withf (0) = 0.
(a) Show that the Fourier series for this function is
f (x)=
∞
k=0
2
k
sin kx,
and f has a jump discontinuity at the origin with
6.14 Exercises 225
f (0+) =
π
2
,f(0−)=−
π
2
, and f (0+) − f (0−)=π.
(b) Show that
max
0≤x≤
π
n
s
n
(x) −
π
2
=
π
0
sin θ
θ
dθ −
π
2
.
(c) The result (b) is a manifestation of the Gibbs phenomenon, that
is, near a jump discontinuity, the Fourier series of f overshoots (or
undershoots) it by approximately 9% of the jump.
(d) If d
n
(x)=s
n
(x) − f (x)in0≤ x ≤ 2π, show that
d
′
n
(x)=2πD
n
(x) ,
where D
n
(x) is given by (6.10.15).
(e) Using D
n
(x)=
n
k=−n
e
ikx
, show that the first critical point of d
n
(x)
to the right of the origin occurs at x
n
= π/
n +
1
2
, and that
lim
n→∞
d
n
(x
n
)=2
π
0
sin θ
θ
dθ − π.
(f) Draw the graph of the fortieth partial sum
s
40
(x)=
40
k=1
2sinkx
k
, −2π<x<2π,
and then examine the Gibbs phenomenon for the function f (x).
19. Consider the characteristic function of the interval [a, b] ⊂ [−π, π]de-
fined by
f (x)=χ
[a,b]
(x)=
⎧
⎨
⎩
1,a≤ x ≤ b
0, otherwise.
Show that the Fourier series in a ≤ x ≤ b is given by
f (x) ∼
1
2π
(b − a)+
k=0
exp (−ika) − exp (−ikb)
2πik
· e
ikx
.
20. (a) Obtain the Fourier series of f (x)=x (π − x), 0 ≤ x ≤ π.
(b) Derive the foll owing the numerical series
1 −
1
3
3
+
1
5
3
−
1
7
3
+ ... =
π
3
12
,
1+
1
3
3
−
1
5
3
−
1
7
3
+
1
9
3
+
1
11
3
− ... =
3
√
2 π
3
128
.
226 6 Fourier Series and Integrals with Applications
21. (a) Show that the Fourier series of the triangular function with vertices
at (0, 0), (π/2, 1) and (π,0) defined by
f (x)=
⎧
⎨
⎩
2x
π
, 0 ≤ x ≤
π
2
2
π
(π − x) ,π/2 ≤ x ≤ π
is
f (x)=
8
π
2
sin x
1
2
−
sin 3x
3
2
+
sin 5x
5
2
−
sin 7x
7
2
+ ...
.
(b) Show that
∞
n=1
1
(2n − 1)
2
=
π
2
8
.
22. Obtain the Fourier sine series and the Fou rier cosine series of the fol -
lowing functions:
(a) f (x)=1, 0<x<a,
(b) f (x)=x,0<x<a,
(c) f (x)=x
2
,0<x<a.
23. Find the full Fourier series of the following functions
(a) f (x)=x,0<x<a,
(b) f (x)=
⎧
⎨
⎩
−1 − x, −1 <x<0,
+1 − x, 0 <x<1.
(c) f (x)=
⎧
⎨
⎩
0, 0 ≤ x ≤ π,
1,π≤ x ≤ 2π.
24. Obtain the Fourier cosine series for the function
f (x)=
⎧
⎨
⎩
cos x, 0 <x≤
π
2
,
0,
π
2
≤ x<π.
25. Show that
1
2i
n=0
1
n
e
inx
is the complex Fourier series of the 2π-periodic
sawtooth function d efined by f (0) = 0, and
6.14 Exercises 227
f (x)=
⎧
⎨
⎩
−
1
2
(π + x) , −π<x<0,
1
2
(π − x) , 0 <x<π.
26. Suppose f (x)andg (x) have the foll owing Fourier series expansion in
−π ≤ x ≤ π:
f (x) ∼
1
2
a
0
+
∞
k=1
(a
k
cos kx + b
k
sin kx) ,
g (x) ∼
1
2
α
0
+
∞
k=1
(α
k
cos kx + β
k
sin kx) ,
where f (x)andg (x) together with their first two derivatives are con-
tinuous on −π ≤ x ≤ π,andf (−π)=f (π), f
′
(−π)=f
′
(π),
g (−π)=g (π), g
′
(−π)=g
′
(π)hold.
Prove that the following general Parseval relation holds:
1
π
π
−π
f (x) g (x) dx =
1
2
a
0
α
0
+
∞
k=1
(a
k
α
k
+ b
k
β
k
) .
When f (x)=g (x), then the Parseval relation (6.5.10) is a special case
of the above result.
27. Obtain the Fourier integral representation for the following functions:
(a) f (x)=H (a −|x|)=
⎧
⎨
⎩
1, |x| <a
0, |x| >a,
(b) f (x)=
⎧
⎨
⎩
sin x, |x| <π
0, |x| >π.
28. If f (x), x ∈ R, is defined by
f (x)=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
−1, −a<x<0,
+1, 0 ≤ x<a,
0, otherwise,
show that f (x) has the Fourier sine integral representation
f (
x)=
2
π
∞
0
1
k
(1 − cos ka)sinkx dk.
228 6 Fourier Series and Integrals with Applications
29. If f (x)=e
−x
,0<x<∞, show that
(a) the Fourier sine integral representation is
f (x)=
2
π
∞
0
k sin kx
1+k
2
dk,
(b) the Fourier cosine integral representation is
f (x)=
2
π
∞
0
cos kx
1+k
2
dk.
30. If f (x) is defined by
f (x)=
⎧
⎨
⎩
0,x<0,
e
−x
,x>0,
show that
(a) the Fourier integral representation of f (x)is
f (x)=
1
π
∞
0
(cos kx + k sin kx)
(1 + k
2
)
dk,
(b) the Fourier cosine integral representation of f (x)is
f (x)=
1
2π
∞
−∞
1 − ik
1+k
2
e
ikx
dk.
31. (a) Obtain both the complex Fourier series and the usual Fourier series
of f (x)=exp[x (1 + 2πi)] on the interval [−1, 1].
(b)Findthesumofeachoftheseries
∞
k=1
1
(1 + π
2
k
2
)
and
∞
k=1
(−1)
k
(1 + π
2
k
2
)
.
32. Use Example 6.7.2 to calculate the value of the following series:
(a)
∞
k=1
1
(4k
2
− 1)
, (b)
∞
k=1
(−1)
k
(4k
2
− 1)
,
(c)
∞
k=1
1
(4k
2
− 1)
2
, an d (d)
∞
k=1
k
2
(4k
2
− 1)
2
.
33. Show that the complex Fourier series of f (x)=x is given by
x ∼
∞
k=1
(−1)
k
k
ie
ikx
+
−∞
k=−1
(−1)
k
k
ie
ikx
.
6.14 Exercises 229
34. (a) Show that the Fourier series for f (x) is defined by
f (x)=
⎧
⎨
⎩
sin 2x, 0 ≤ x ≤
π
2
0,
π
2
≤ x ≤ π,
is
f (x)=
1
π
+
1
2
sin 2x −
2
π
∞
k=1
cos 4kx
(4k
2
− 1)
.
(b) Show that
∞
k=1
1
(4k
2
− 1)
2
=
1
16
π
2
− 8
.
(c) Find the sum of the infinite series
sin (4x)
1.2.3
+
sin (2.4x)
3.4.5
+
sin (3.4x)
5.6.7
+ ..., 0 ≤ x ≤ π.
35. (a) Obtain the complex Fourier series of
f (x)=cos(ax) , −π ≤ x ≤ π,
where a is real but not an integer.
(b) Hence, show that
π cot πx =
1
x
−
∞
k=1
2x
(k
2
− x
2
)
.
(c) Derive the product formula
sin πx = πx
∞
3
n=1
1 −
x
2
n
2
.
(d) Show that
π
2
=
∞
3
n=1
2n
(2n − 1)
·
2n
(2n +1)
=
2
1
·
2
3
·
4
3
·
4
5
·
6
5
·
6
7
·
8
7
·
8
9
....
36. Obtain the Fourier series of the following functions:
(a) f (x)=e
x
, 0 ≤ x ≤ 2π, f (x +2π )=f (x).
(b) f (x)=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
+1, −π<x<−
π
2
, 0 <x<
π
2
,
−1, −
π
2
<x<0,
π
2
<x<π,
0,x=
nπ
2
,n=0, +
1, + 2,...
230 6 Fourier Series and Integrals with Applications
and draw the graph of this function.
(c) f (x)=x − [x],
where [x] is the greatest integer not exceeding x.
37. Find the Fourier series for each of the functions f (x)in−l<x<land
f (x) is defined outside this interval so that f (x +2l)=f (x) for all x:
(a) f (x)=
⎧
⎨
⎩
0, −l<x<0,
l, 0 <x<l.
(b) f (x)=
⎧
⎨
⎩
−x, −l ≤ x<0,
x, 0 ≤ x<l.
(c) f (x)=
⎧
⎨
⎩
l + x, −l ≤ x<0,
l − x, 0 ≤ x<l.
(d)
f (x)=
⎧
⎨
⎩
0, −l ≤ x<0,
x
2
, 0 ≤ x<l.
Examine th e Gibbs phenomenon at the points of discontinuity at x =0
and x = l for the function in (a).
38. Prove the following identities:
(a)
1
2
+
n
k=1
cos kx =
sin
n +
1
2
x
2sin
x
2
.
(b) s
n
(x)=
1
π
π
−π
f (ξ + x)
sin
n +
1
2
ξ
2sin
1
2
ξ
dξ,
where s
n
(x)isthenth partial sum of a Fourier series of f (x)in(−π, π).