C.1 FINAL VALUE THEOREM
A useful property of the Laplace transform allows us to determine the value of the function yðtÞ knowing only
the Laplace transform of the function. The final value theorem states
lim
t!1
yðtÞ ¼ lim
s!1
½sYðsފ
ðC:6Þ
Let’s apply this to the unit step function
lim
s!0
sYðsÞ ¼ lim
s!0
s
1
s
¼ 1
ðC:7Þ
Clearly this is the correct limit!
C.2 LAPLACE TRANSFORM PAIRS
Table C.1 provides Laplace transform pairs for a variety of important engineering functions. More extensive lists
and methods for more complex transformations may be found in most advanced engineering mathematics texts.
Reference
1. Schiff, J. L., The Laplace Transform: Theory and Applications, Springer, New York, 1999.
Table C.1 Laplace Transform Pairs
f ðtÞ
L ð f Þ
1
1
1
s
2
t
1
s 2
3
t
2
2!
s 3
4
t
n
n!
s nþ1
where n is integer
5
e
Àat
1
s þ a
6
te
Àat
1
ðs þ aÞ
2
6
e
Àat
À e
Àbt
b À a
1
ðs þ aÞðs þ bÞ
7
sin at
a
s 2 þ a 2
8
cos at
s
s 2 þ a 2
Appendix C Laplace Transform Basics 577