Page 125 - Circuit Analysis II with MATLAB Applications

Properties of the Laplace Transform

Proof:

We express the integral of (4.23) as two integrals, that is,

t 0 t
³ f W dW = ³ f W dW + ³ f W dW (4.24)
– f – f 0

The first integral on the right side of (4.24), represents a constant value since neither the upper, nor
the lower limits of integration are functions of time, and this constant is an initial condition denoted

as f 0 . We will find the Laplace transform of this constant, the transform of the second integral
on the right side of (4.24), and will prove (4.23) by the linearity property. Thus,

f f e – st f
L f0 ^ ` = ³ f0 e – st dt = f 0 ³ e – st dt = f 0 --------
0 0 s – 0 (4.25)

·
f0 = u 0 – – f0 § ------------ = f0
------------
© s ¹ s
This is the value of the first integral in (4.24). Next, we will show that
t Fs
³ f W dW œ -----------
s
0
We let
t
gt = ³ f W dW
0
then,
g' t = f W
and
0
g0 = ³ f W dW = 0
0
Now,

`
^
L g' t ` = Gs = sL gt – g0 = Gs – 0
^
sL gt ` = Gs
^
-----------
L gt ` = Gs
^
s
­ t ½ Fs
L ® ³ f W dW ¾ = ----------- (4.26)
¯ 0 ¿ s

and the proof of (4.23) follows from (4.25) and (4.26).

Circuit Analysis II with MATLAB Applications 4-7
Orchard Publications