A right sided signal's *initial value*
and *final value*
(if finite) can be found from its Laplace transform
by the following theorems:

**Initial value theorem:**

**Final value theorem:**

**Proof:** As
for , we have

- When
, the above equation becomes

i.e.,

- When
, we have

i.e.,

- If there are poles on the right side of the S-plane, will contain exponentially growing terms and therefore is not bounded, does not exist.
- If there are pairs of complex conjugate poles on the imaginary axis, will contain sinusoidal components and is not defined.
- If there are poles on the left side of the S-plane, will contain exponentially decaying terms without contribution to the final value.
- Only when there are poles at the origin of the S-plane, will contain constant (DC) component which is the final value, the steady state of the signal.

where are the poles, and by assumption. The corresponding signal in time domain:

All terms except the first one represent exponentially decaying/growing or sinusoidal components of the signal. Multiplying both sides of the equation for by and letting , we get:

We see that all terms become zero, except the first term . If all poles are on the left side of the S-plane, their corresponding signal components in time domain will decay to zero, leaving only the first term , the final value .

**Example 1:**

First find :

When , we get . Next we apply the final value theorem:

**Example 2:**

According to the final value theorem, we have

However, as the inverse Laplace transform

is unbounded (the first term grows exponentially), final value does not exist.

The final value theorem can also be used to find the DC gain of the system,
the ratio between the output and input in steady state when all transient
components have decayed. We assume the input is a unit step function ,
and find the final value, the steady state of the output, as the DC gain of the
system:

**Example 3:**

The DC gain at the steady state when can be found as