±
²
±
3.
Laplace
Transform
3A.
Elementary
Properties
and
Formulas
1
3A1.
Show
from
the
deFnition
of
Laplace
transform
that
L
(
t
)
=
s
2
,
s
>
0
.
3A2.
Derive
the
formulas
for
L
(
e
at
cos
bt
)
and
L
(
e
at
sin
bt
)
by
assuming
the
formula
1
L
(
e
t
)
=
s
−
is
also
valid
when
is
a
complex
number;
you
will
also
need
L
(
u
+
iv
)
=
L
(
u
)
+
i
L
(
v
)
,
for
a
complexvalued
function
u
(
t
)
+
iv
(
t
).
3A3.
±ind
−
1
F
(
s
)
for
each
of
the
following,
by
using
the
Laplace
transform
formulas.
L
(±or
(c)
and
(e)
use
a
partial
fractions
decomposition.)
1
3
1
1
+
2
s
1
a)
1
b)
c)
d)
e)
s
2
s
+
3
s
2
+
4
−
4
s
3
s
4
−
9
s
2
2
3A4.
Deduce
the
formula
for
L
(sin
at
)
from
the
deFnition
of
Laplace
transform
and
the
formula
for
L
(cos
at
),
by
using
integration
by
parts.
3A5.
a)
±ind
L
(cos
2
at
)
and
L
(sin
2
at
)
by
using
a
trigonometric
identity
to
change
the
form
of
each
of
these
functions.
b)
Check
your
answers
to
part
(a)
by
calculating
L
(cos
2
at
)
+
L
(sin
2
at
).
By
inspection,
what
should
the
answer
be?
³
⎨
1
±
3A6.
a)
Show
that
=
,
s
>
0
,
by
using
the
wellknown
integral
L
∗
t
s
⎪
2
∗
±
−
x
e
dx
=
.
2
0
(Hint:
Write
down
the
deFnition
of
the
Laplace
transform,
and
make
a
change
of
variable
in
the
integral
to
make
it
look
like
the
one
just
given.
Throughout
this
change
of
variable,
s
behaves
like
a
constant.)
b)
Deduce
from
the
above
formula
that
L
(
∗
t
)
=
∗
±
,
s
>
0
.
2
s
3
/
2
2
3A7.
Prove
that
L
(
e
t
)
does
not
exist
for
any
interval
of
the
form
s
>
a
.
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 Fall '09
 vogan
 Formulas, Laplace

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