Laplace Transform Properties
From ControlTheoryPro.com

1 Properties and theorems
Given the functions f(t) and g(t), and their respective Laplace transforms F(s) and G(s):
the following table is a list of properties of unilateral Laplace transform:
Time domain  Frequency domain  Comment  

Linearity  Can be proved using basic rules of integration.  
Frequency differentiation  
Frequency differentiation  More general form  
Differentiation  Write the exact integral form of the given function, and add another integral to complement the former to deduce the sum to indefinite integration of a differential. Next few steps are simple.  
Second Differentiation  Apply the Differentiation property to .  
General Differentiation  Follow the process briefed for the Second Differentiation.  
Frequency integration  
Integration  is the Heaviside step function.  
Scaling  
Frequency shifting  
Time shifting  is the Heaviside step function  
Convolution  
Periodic Function  is a periodic function of period so that 
 Initial value theorem:
 Final value theorem:
 , all poles in lefthand plane.
 The final value theorem is useful because it gives the longterm behaviour without having to perform partial fraction decompositions or other difficult algebra. If a function's poles are in the right hand plane (e.g. or ) the behaviour of this formula is undefined.
1.1 Proof of the Laplace transform of a function's derivative
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows:
 (by parts)

yielding
and in the bilateral case, we have
