Page 503 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 503
B
Introduction
to Laplace
Transformation
Definition
If fit) is a known function of t for values of t > 0, its Laplace transform (LT),
f(s), is defined by the equation
f ( s ) = = (B-l)
where 5 is a complex variable. The integral exists for the real part of > 0
provided fit) is an absolutely integrable function of t in the time interval 0
to 00.
Example B-l
Let fit) be a constant c for t > 0. Its LT is
_ C
flc = / ce dt = -
0 ~ S
which exists for Ris) > 0.
Example B-2
Let fit) = t. Its LT is found by integration by parts, letting
u = t du = dt
dv = e '' dt V = - ■
The result is
te~
f l t = - + - r e "'dt = \ R(s)>
■A)
s
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