GEM 601 Advanced Engineering Mathematics 1
First Semester, AY 2017-2018
EXERCISES: Fourier Series
1. A real-valued function f is said to be periodic with period 2 p if f  x   f  x  2 p  . For example,
4 is a period of sin x since sin x  sin  x  4  . The smallest value of 2 p for which
f  x   f  x  2 p  holds is called the fundamental period of f . For example, the fundamental
period of f  x   sin x is 2 p  2 .
What is then the fundamental period of each of the following functions?
b) f  x   sin
4
x
L
e) f  x   sin 3 x  cos 2 x
a) f  x   tan x
d) f  x   2
2. Determine whether the function is even, odd, or neither.
a) f  x   x cos x
b) f  x   x 3  4 x
c) f  x   sin x  sin 2 x
c) f  x   e x  e  x
 x  5,  2  x  0
e) f  x   x 5
 x  5 , 0  x  2
d) f  x   
3. Find the Fourier expansion of the function whose definition over one period is
t
f t   
0
0  t 1
1 t  2
4. For c units of time water flows through a turbine at a constant rate b . During the next 2  c  units
of time the constant rate of flow is a . Then for c more units of time the flow rate is b again.
Thereafter, water discharges through the turbine in the same way periodically. Find a half-range
cosine expansion for the periodic rate r t  at which water flows through the turbine.
5. During the two-second interval 0  t  2 a voltage E t   2t  t 2 is impressed upon an electric
circuit. Find the half-range sine expansion of E .
6. Suppose a uniform beam of length L is simply supported at x  0 and at x  L . If the load per unit
length is given by w x  
w0 x
, 0  x  L , then the differential equation for the deflection y x  is
L
d4y
EI 4  wx 
dx
where E , I , and wo are constants.
a) Expand w x  in a half-range sine series.
b) Use the method of undetermined coefficients to find a particular solution y  x  of the differential
equation.