Figure 12-8. The Laplace transform operator.
In the defining equation (12.175) the parameter sis selected such that the in-
tegral exists and many times it is expressed as a complex variable s=σ+i ω where
σ and ωare real and iis an imaginary component satisfying i^2 =− 1. The Laplace
transform can be studied with or without employing knowledge of complex variables.
Example 12-11. (Laplace transform)
Find the Laplace transform of sin(α t)where αis a nonzero constant.
Solution
By definition L{ sin(α t); t→s}=
∫∞0sin(α t)e−st dtIntegrate by parts with u= sin(α t)and dv =e−st dt to obtain
I=∫∞0sin(α t)e−st dt =−sin(α t)e−st
s∞
0+α
s∫∞0cos(α t)e−stdtIntegrate by parts again with u= cos(α t)and dv =e−st dt and show
I=∫∞0sin(α t)e−st dt =α
s[
cos(α t)e−st
−s∞
0−α
sI]This last equation simplifies to
(1 +α^2
s^2 )I=αs^2 or I=
∫∞0sin(α t)e−st dt =L{ sin(α t); t→s}=α
s^2 +α^2