Let 
be the Laplace transform of a function 
. The final value theorem states,
under its stability conditions, that
 |
A sufficient condition is that every pole of 
lie in the open left half-plane, except possibly for a simple pole at 
. The pole condition is essential:
the right-hand limit may exist even when 
oscillates and has no final value.
For a sequence 
with Z-transform 
, the corresponding discrete form is
 |
with the analogous requirement that the poles of 
lie inside the unit circle.
