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)
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 f
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 de
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 L
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) :
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 sho
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 fo
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
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)
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 give