What are the properties of Laplace Transform?

Laplace Transform is a method used for solving differential equations. Firstly, The time domain differential equations are transformed into algebraic equations of the frequency domain. In this we firstly solve the equations in frequency domain. Then the result obtained is transformed in time domain. This then provides us with the unique solution of the differential equation. Therefore, the Laplace Transform is

The advantages are:

• It is systematic.
• It gives a total solution (transient and sustained solution) in one operation.
• The initial conditions are automatically specified in the transformed equations.

There are many functions which do not have Laplace Transforms. These functions are not generally used in analysis of linear systems. But some conditions can be defined to get Laplace transformation of such functions. The Dirichlet condition defines the necessary condition for transformation of some functions such as:

• Firstly, the function should be continuous.
• Secondly the function should be a single-valued.
• Lastly, the function must be of exponential order.

Properties of Laplace Transform:

1)    Linearity Property:

If f₁(t) and f₂(t) are two functions of time then in domain of convergence.

2)    Differentiation Property:

If x(t) is function of time then Laplace transform of n^th derivative is

3) Integration Property:

The integral of Laplace of n^th order integral is

4) Time Shifting Property

5) Shifting in S-Domain

Applications

• For solving electrical circuits.
• In Digital Signal Processing.
• To study transient and steady state response.
• Makes the circuit calculations simpler.
• LTI systems can be characterized by Laplace Transform.
• We use Laplace Transform to study the behavior of control system.

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