ABSTRACT
The Laplace transform is linear:
+ = +[ ( ) ( )] [ ( )] [ ( )].1 1 2 2 1 1 2 2L C f x C f x C L f x C L f x (6.3)
This follows from the linearity of the integral (1.1). The inverse transform is also linear:
+ = + − − −L C f p C f p C L f p C L f p[ ˆ ( ) ˆ ( )] [ ˆ ( )] [ ˆ ( )].1 1 1 2 2 1 1 1 2 1 2 (6.4)
Property 2
Let =[ ( )] ˆ( ).L f x f p Then
=
[ ( )] 1 ˆ .L f kx k f p
k (6.5)
This can be proven using the change of a variable, x = t/k:
( ) 1 ( ) 1 ˆ .
