Exercises: Complex Integration
2.3
2002/02/20
2002/07/19
Steven
Cox
cox@rice.edu
Charlet
Reedstrom
charlet@rice.edu
Jacob
Grabczewski
jgrab@owlnet.rice.edu
Steven
Cox
cox@rice.edu
complex integration
inverse Laplace transform
Laplace transform
(Blank Abstract)
- Let us confirm the representation of this Cauchy's Theorem equation in the
matrix case. More precisely, if
Φ
z
z
I
B
is the transfer function associated with
B then this Cauchy's Theorem equation
states that
Φ
z
j
1
h
k
1
d
j
Φ
j
,
k
z
λ
j
k
where
Φ
j
,
k
1
2
z
C
j
Φ
z
z
λ
j
k
1
Compute the
Φ
j
,
k
per for the B in
this
equation from the discussion of Complex
Differentiation. Confirm that they agree with those
appearing in this equation from the Complex
Differentiation discussion.
-
Use this inverse Laplace Transform equation to
compute the inverse Laplace transform of
1
s
2
2
s
2
.
-
Use the result of the previous exercise to solve, via the
Laplace transform, the differential equation
t
x
t
x
t
t
t
,
x
0
0
Hint: Take the Laplace transform of each side.
- Explain how one gets from
r
1
and
p
1
to
x
1
t
.
- Compute, as in
fib4.m, the residues
of
ℒ
x
2
s
and
ℒ
x
3
s
and confirm that they give rise to the
x
2
t
and
x
3
t
you derived in the discussion of Chapter 1.