Environmental Engineering Reference
In-Depth Information
Fig. B.1
A half circle
C
=
L
+
C
R
β
+
i∞
n
j
=
1
Res
f
(
s
)
e
st
s
j
.
1
f
e
st
d
s
f
(
t
)=
(
s
)
=
,
(B.25)
2
π
i
β
−
i
∞
Corollary.
Consider the irreducible rational function
f
A
(
s
)
(
s
)=
where the degree
B
(
s
)
n
of
B
(
s
)
is higher than that of
A
(
s
)
.Wehave:
1. When
B
(
s
)
has
n
distinct zero points
s
j
of order 1,
j
=
1
,
2
, ··· ,
n
,
n
j
=
1
s
j
)
B
(
A
(
e
s
j
t
f
(
t
)=
t
>
0
.
(B.26)
s
j
)
2. When
B
(
s
)
has the zero point
s
1
of order
m
and zero points
s
m
+
1
,
s
m
+
2
,
···
,
s
n
of
order 1,
d
s
m
−
1
e
st
n
∑
j
=
m
+
1
d
m
−
1
s
j
)
B
(
A
(
1
m
A
(
s
)
e
s
j
t
f
(
t
)=
+
lim
s
(
s
−
s
1
)
,
t
>
0
.
s
j
)
(
m
−
1
)
!
B
(
s
)
→
s
1
(B.27)
These two equations together are called the
Heaviside expansion
andplayanimpor-
tant role in solving equations using the Laplace transformation.
In calculating lim
s
d
s
m
−
1
e
st
in Eq. (B.27), common factors in
d
m
−
1
m
A
(
s
)
B
(
s
−
s
1
)
(
s
)
→
s
1
are reducible so that
e
st
becomes an analytical
m
A
(
s
)
m
and
B
(
s
−
s
1
)
(
s
)
(
s
−
s
1
)
B
(
s
)
function. Note that the (
m
−
1
)
-th derivative of an analytical function is still an ana-
e
st
is thus reduced to the problem
of finding the value of an analytical function at
s
1
.
d
s
m
−
1
d
m
−
1
m
A
(
s
)
lytical function. To find lim
s
(
s
−
s
1
)
B
(
s
)
→
s
1
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