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ELEC ENG 1 PROBLEM SHEET 2:
FOURIER SERIES
1. A periodic function f(t) is defined by f (t) = t2 + t, −π < t < π, f (t + 2π) =
f (t). Sketch the graph of the function for −3π < t < 3π and find the Fourier
series expansion of the function.
2 A function f(t) is periodic with period 2π is defined in the interval (0, 2π) by:
f = t, 0 ≤ t ≤ π/2
f = π/2, π/2 ≤ t ≤ π
f = π − t/2, π ≤ t ≤ 2π.
Sketch the graph of the function in (-2π, 3π) and find its Fourier series expansion.
3. A 2π−periodic function f is defined in the interval (-π, π) by f = t. Sketch
the graph of the function and show that
its Fourier series is given by
f=
Deduce that
∞
X
(−1)n+1
n2
=
∞
X
π2
(−1)n n−2 cos nt
+4
3
n=1
π2
12 .
n=1
4. Suppose that g(t), h(t) are 2π periodic functions defined in (-π, π) by g =
t2 , h = t. What is the Fourier series of the function f (t) = g(t) + h(t)?
5.Give reasons why the functions: (a)1/(3 − t), (b)sin(1/(2 − t))do not satisfy
Dirichlet’s conditions in (0,2π).
6. The following function is periodic with period 2π. Sketch the function in
(-4π, 4π) and obtain its Fourier series.
π
2
f = 2 cos t, −π/2 ≤ t ≤ π/2
f = 0, π/2 ≤ t ≤ π.
f = 0, −π ≤ t ≤ −
1
7. A function f(t) has period 2 and is defined in (0,2) by
f = 3t, 0 < t < 1
f = 3, 1 < t < 2
Sketch f(t) in (-4,4) and find its Fourier series expansion.
8. Find the Fourier series expansion of the function f (t) = t2 for −T < t <
T, f (t + 2T ) = f (t). Sketch the function in (-3T,3T).
9. In the interval (0, 4) the function f (t) = t. Find a half-range cosine series
and a half-range sine series for the function. Sketch the functions represented
by the two functions in (-20,20).
10. Find the Fourier series of the function defined by
f = 1, −1 < t < 0
f = cos πt, 0 < t < 1.
f (t + 2) = f (t).
To what value does the series converge when t=1?
11. Use the result of Q3 to show by integration that in (-π, π) we may write
∞
X
(−1)n
t3 − π 2 t = 12
n2 sin nt.
n=1
12. Use the result of Q3 to show by differentiation that in (-π, π) we may write
∞
X
(−1)n+1
t=2
sin nt.
n
n=1
13. Find the complex form of the Fourier series for the function f(t) defined by
f = cos t/2, −π < t < π
f (t + 2π) = f (t).
Sketch the function.
14. By applying Parsevals theorem to the function f (t) =
∞
X
1
π2
2T ) = f (t) show that
n2 = 6 .
n=1
2
2t
T ,0
< t < T, f (t +