MS 210 Homework Set (4)
(Trigonometric Fourier Series)
QUESTION 1: Show that the following functions have the given trigonometric
Fourier series and sketch the periodic extension of each function.
(i) f : [, ] R : x
x+
x
4
+
2
(ii) f : [, ] R : x |x|
if
MS 210 Homework Set (7)
Begin with The Fourier Integral Formula written in the form
f (t) = lim
c 0
c
cfw_ A( ) cos t + B ( ) sin t d
where
A( ) =
1
f () cos d
and
B ( ) =
1
f () sin d
.
DEFINE the function F [f ] : R R : F [f ]( ) by
[ A( ) i B ( ) ]
2
MS 210 Homework Set (8)
QUESTION 1
Let f : R R : t f (t). Assume that f , together with all of its derivatives,
is piecewise continuous and absolutely integrable. In particular, f (t) 0 and its
nth derivative f (n) (t) 0 as t . As usual, denote the Fourie
MS 210 Homework Set (9)
QUESTION 1
Calculate the Laplace Transform, L[f ](s), of the function
f : [0, ) R : t
Solution: L[f ](s) =
1
0
if
t [0, 1]
if
t [0, 1]
1 es
.
s
QUESTION 2
Using the Table of Laplace Transforms, or otherwise, calculate the followin
MS 210 Homework Set (6)
REMARKS:
Consider a complex-valued function f which is dened on a real interval [a, b]. By
decomposing f into its real and imaginary parts we can write
f : [a, b] C : t f (t) := u(t) + i v (t)
where the real-valued functions u and
MS 210 Homework Set (5)
QUESTION 1
Use any Theorem or Test you wish (provided you state clearly the results you use)
to explain why
cos t
dt
t2
1
converges.
QUESTION 2
Use integration by parts to show that
sin t
cos t
dt =
t
t
cos t
dt.
t2
QUESTION 3
Us
MS 210 Homework Set (1)
QUESTION 1
Write down a rigorous proof (i.e. using > 0 and N N ) of the following result:
The Sandwich Theorem : Let the three sequences an , bn and cn be such that
an bn cn for all n N . If both
an L and cn L as n
then bn L as n
MS 210 Homework Set (2)
QUESTION 1
In each of the following, nd the (pointwise) limit of the given sequence of functions
and state, with justication, whether the convergence is uniform or not:
(i) fn : [0, 1] R : x fn (x) := xn (1 x).
nx
.
1 + n2 x2
nx
(i
MS 210 Homework Set (3)
(Complete Sets Of Functions)
QUESTION 1
Compute all the Bernstein polynomials of each of the three functions
f, g, h : [0, 1] R which are dened by f (x) = 1, g (x) = x and h(x) = x2 .
Hint: Use Lemma 1.
QUESTION 2
Consider the vect
MS 210 Homework Set (10)
QUESTION 1
In the case of each of the following dierential equations, nd the roots of the corresponding characteristic equation, and hence, write down the solution of the given
dierential equation:
(a)
d2
d
x(t) + x(t) 6x(t) = 0
2