In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.
It is also known under the abbreviation IVT.
Let
F
(
s
)
=
∫
0
∞
f
(
t
)
e
−
s
t
d
t
be the (one-sided) Laplace transform of ƒ(t). The initial value theorem then says
lim
t
→
0
f
(
t
)
=
lim
s
→
∞
s
F
(
s
)
.
Based on the definition of Laplace transform of derivative we have:
s
F
(
s
)
=
f
(
0
−
)
+
∫
t
=
0
−
∞
e
−
s
t
f
′
(
t
)
d
t
thus:
lim
s
→
∞
s
F
(
s
)
=
lim
s
→
∞
[
f
(
0
−
)
+
∫
t
=
0
−
∞
e
−
s
t
f
′
(
t
)
d
t
]
But
lim
s
→
∞
e
−
s
t
is indeterminate between t=0− to t=0+; to avoid this, the integration can be performed in two intervals:
lim
s
→
∞
[
∫
t
=
0
−
∞
e
−
s
t
f
′
(
t
)
d
t
]
=
lim
s
→
∞
{
lim
ϵ
→
0
+
[
∫
t
=
0
−
ϵ
e
−
s
t
f
′
(
t
)
d
t
]
+
lim
ϵ
→
0
+
[
∫
t
=
ϵ
∞
e
−
s
t
f
′
(
t
)
d
t
]
}
In the first expression,
e
−
s
t
=
1
w
h
e
n
0
−
<
t
<
0
+
.
In the second expression, the order of integration and limit-taking can be changed. Also
lim
s
→
∞
e
−
s
t
(
t
)
=
0
w
h
e
r
e
0
+
<
t
<
∞
.
Therefore:
lim
s
→
∞
[
∫
t
=
0
−
∞
e
−
s
t
f
′
(
t
)
d
t
]
=
lim
s
→
∞
{
lim
ϵ
→
0
+
[
∫
t
=
0
−
ϵ
f
′
(
t
)
d
t
]
}
+
lim
ϵ
→
0
+
{
∫
t
=
ϵ
∞
lim
s
→
∞
[
e
−
s
t
f
′
(
t
)
d
t
]
}
=
f
(
t
)
|
t
=
0
−
t
=
0
+
+
0
=
f
(
0
+
)
−
f
(
0
−
)
+
0
By substitution of this result in the main equation we get:
lim
s
→
∞
s
F
(
s
)
=
f
(
0
−
)
+
f
(
0
+
)
−
f
(
0
−
)
=
f
(
0
+
)
