Fourier analysis (MMG710/TMA362)
Time: 2012-01-07, 08:30–12:30
Tools: Only the attached sheet of formulas. No calculator or handbook is allowed.
Questions: Ragnar Freij, 0703-088304.
Grades: Each problem gives 4 points. For MMG710 grades are G (12-17 points) and VG (18-24 points).
For TMA362 grades are 3 (12-14 points), 4 (15-17 points) and 5 (18-24 points).
1 Use Fourier transform to compute the integral
Z ∞
−∞
(x2
1
dx.
+ 2x + 2)2
2 Solve the boundary value problem
u0t = 2u00xx ,
t > 0,
u(0, t) = u(π, t) = 0,
3 Let
0 < x < π,
u(x, 0) = cos(3x).
(
t, 0 < t < 1,
f (t) =
1, t > 1.
Use Laplace transform to solve the initial value problem
y 0 + 2y = f,
y(0) = 0.
4 Formulate and prove Bessel’s inequality (any version is fine).
5 Let cn be the coefficients in the Fourier series
2
ex =
∞
X
cn einx ,
0 < x < 2π.
n=−∞
Is it true or false that
x2
2xe
=
∞
X
incn einx ,
0 < x < 2π?
n=−∞
Motivate your answer carefully. If you use a known theorem, formulate it precisely and explain why
all conditions in the theorem hold.
6 Find the complex Fourier series of the function (cos x)n , where n is a positive integer. Use the result
to compute the sum
n 2
X
n
k
k=0
(where nk = n!/k! (n − k)!).
Good luck!
Hjalmar
Some formulas in Fourier analysis
Trigonometric identities
eix − e−ix
eix + e−ix
,
sin x =
,
2
2i
cos(x + y) = cos x cos y − sin x sin y,
sin(x + y) = sin x cos y + cos x sin y,
1 − cos 2x
1 + cos 2x
sin2 x =
,
cos2 x =
,
2
2
cos(x − y) − cos(x + y)
cos(x − y) + cos(x + y)
sin x sin y =
,
cos x cos y =
,
2
2
sin(x + y) + sin(x − y)
sin x cos y =
.
2
eix = cos x + i sin x,
cos x =
Hyperbolic functions
cosh x =
ex + e−x
,
2
sinh x =
ex − e−x
.
2
Laplace transforms
f (t)
F (s)
f (t)
F (s)
tk
k!
eat
1
s−a
sk+1
f (at)
F (s/a)/a
sin(at)
a
a2 + s2
ect f (t)
F (s − c)
cos(at)
s
a2 + s2
f (t − a)H(t − a)
e−as F (s)
(n)
Pfn (t)
s F (s) − j=1 sn−j f (j−1) (0)
n
Fourier transforms
e−x
f (x)
fˆ(ξ)
√
2
/2
2π e−ξ
2
/2
1
x2 + 1
πe−|ξ|
e−|x|
2
ξ2 + 1
χ(x)
2 sin ξ
ξ
sin x
x
πχ(ξ)
(where χ(x) equals 1 for |x| < 1 and 0 else)
f (x)
fˆ(ξ)
f (x − c)
e−icξ fˆ(ξ)
eicx f (x)
fˆ(ξ − c)
f (ax)
fˆ(ξ/a)/a
f 0 (x)
iξ fˆ(ξ)
xf (x)
i(fˆ)0 (ξ)
fˆ(x)
2πf (−ξ)
tf (t)
−F 0 (s)