Appendix C
Laplace Transform Basics
We offer the following primer of Laplace transforms for review purposes to meet the needs of this text.
Extensive treatment of this subject is available elsewhere [1]. The Laplace transform of a function yðtÞ is
defined as
YðsÞ ¼ L½Yðtފ ¼
ð 1
0
yðtÞe
Àst dt
ðC:1Þ
There are many functions that are useful in modeling linear systems that have analytical forms for their
Laplace transforms. Consider the unit step function, defined by
UðtÞ ¼
0 t < 0
1 t > 0
ðC:2Þ
Applying the definition of the Laplace transform yields
L½Uðtފ ¼
ð 1
0
e
Àst dt ¼
1
s
ðC:3Þ
The unit step function and its Laplace transform form a Laplace transform pair. Table C.1 provides a list of
common Laplace transform pairs.
One of the most important properties of the Laplace transform results from the application of the Laplace
transform to a derivative, as
L
dyðtÞ
dt
¼ sYðsÞ À yð0Þ
ð C:4Þ
Differentiation in the time domain is the same as multiplication in the Laplace transform domain. This
relationship is derived through application of integration by parts of the definition of the Laplace transform,
and can be generalized as
L
d
n yðtÞ
dt n
¼ s
n YðsÞ À s
nÀ1 yð0Þ À s
nÀ2 dyð0Þ
dt
À Á Á Á À
d
nÀ1 yð0Þ
dt nÀ1
ðC:5Þ
576