1
MA360
Handout 4a (week of Jan 30 )
Sp2006 Kallfelz
Note
: New (absolute) due date for Assignment I (See posted assignment sheet for details)
Assignment: pp. 5255 Sheng: 2a), b), e), 3b), c), 4a), d)
Summary of Theorems (Ch 1, Sheng)
Theorems
Description
1.
L
{
af
(
t
) +
bg
(
t
)} =
aL
{
f
(
t
)} +
bL
{g(
t
)}
Corrolat 1a.) For any piecewise continuous
f
:
lim
s
→∞
L
{
f
(
t
)} = 0
Linearity and asymptotic property of LT
2 .:
L
{
D
t
f
(
t
)} =
sL
{
f
(
t
)} –
f
(0)
2a. Corrolary to 2 (not found in Sheng):
L
{
D
t
n
f
(
t
)} =
s
n
L
{
f
(
t
)} –
∑
n
j=
1
s
nj
f
(
j
1)
(0)
4 :
L
{
D
t
1
f
(
t
)} =
1
/
sL
{
f
(
t
)}
2. & 4. give formulae in terms of the LT of
f
(
i.e.,
L
{
f
(
t
)} =
F
(
s
) ) when you’re faced with
computing the LT for the derivative of
f
(or in
the case of Thm 2a, the
n
th derivative) or the
antiderivative of
f
(
D
t
1
{
f
(
t
)}
≡
∫
t
0
f
(
ω
)
d
3.
L
{
f
(
t
)}
=
F
(
s
)
⇒
L
{
f
(
at
)}
=
1
/
a
F
(
s
/
a
)
An obviously useful simplification formula, but
it also tells you that the LT is
not
linear with
respect to the
variable
arguments
of
f.
LT
is
linear with respect to
its
functional
arguments
1
!
5.
L
{
t
n
f
(
t
)} = (1)
n
D
n
s
L
{
f
(
t
)} for
n
= 0,1,2,…
Again, another obviously useful simplification
formula
5a. Corrolary (Example 124)
{ }
( )
1
1
+
+
Γ
=
p
p
s
p
t
L
,
for
any
p
> 1
(From Handout 1c, Formula 1c.1)
The formula
comes especially in handy of integer and half
integer exponents, given Properties 14 as of
the Gamma function as summarized in Handout
1c
6.
( )
{ } ( )
d
F
t
f
t
L
s
∫
∞

=
1
where:
L
{
f
(
t
)}
=
F
(
s
)
Discussed here (Handout 2).
Calculus analogy:
Recall the powerrule for integration did
not
hold for
t
p
when
p
= 1?
(You had to introduce a rigorous defn
of logarithm)
7.
L
{
f
(
t
)}
=
F
(
s
)
⇒
L
{
e
at
f
(
t
)}
=
F
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 Spring '06
 WILLIAMM.KALLFELZ
 lim, Mathematics in medieval Islam, −s, sheng

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