ABSTRACT
Equating the real and imaginary components of Equation (7.7) and Equation (7.8) gives the Laplace transforms for sine and cosine:
L ω =
+ ω ( ) 2 2cos t
s s
(7.9)
L ω =
ω
+ ω ( ) 2 2sin t s
(7.10)
Generalizing to the case when the exponent has both real and imaginary parts,
L
L L
=
− + ω
=
− − ω
=
−
− + ω +
ω
− + ω
= ω + ω
+ ω( ) 1 ( )
1 ( )
( ) ( ) ( cos ) ( sin )
e s a j
s a j s a
s a j
s a e t j e t
We may observe from this that multiplication of the function ( )f t e j t= ω by e at in the time domain produces an s shift in the Laplace domain. is may be generically stated as
L = −( ( )) ( )e f t F s a at (7.11)
7.2.6 Powers of t Let
( ) 1f t t n= +
en,
t t e dtn n st
206
To solve, integrate by parts. Let 1u t n= + and dv e dtst= − . en, ( 1)du n t dtn= +
and = − −
v e s
. Applying udv uv vdu∫ = − ∫ , we get
L∫ ∫= − + + = ++ −∞ + −
1 1 ( )1
t e dt t e s
n s
t e dt n s
From Equation (7.12), we can see that
L L
L L
L L L
=
=
−
= = =
( ) ( )
( ) 1 ( )
( ) 1 ( ) 1 (1) 1
t n s
t
t n
s t
t s
t s s
(7.13)
7.2.7 Delta Function e delta function was formalized by physicist Paul Dirac and is de›ned as follows:
( ) for 0
0 otherwise t tδ = ∞ =
and
( ) 1 0
t dt∫ δ =∞
Although we cannot physically realize δ(τ) in an actual circuit (though it is often approximated), it has many useful applications in circuit theory, and its Laplace transform is
L ∫δ = δ = =−∞ − ∞( ( )) ( ) 1 0
Example 7.1
Find the Laplace transform of the pulse p(t) as shown in Figure 7.1, where A is the amplitude, T1 is the turn-on time, and T2 is the turn-off time.
