### Properties of Laplace Transform

Number of important properties of the Laplace transform are discussed in this section. The table of Laplace transform pairs is developed using these properties.
1. Linearity
The transform of a finite sum of time functions is the sum of the Laplace transforms of the individual functions.
So if F1(s), F2(s), …….. , Fn(s) are the Laplace transforms of the time functions f1(t), f2(t), …… , fn(t) respectively then,

The property  can be further extended if the time functions are multiplied by the constants i.e.

Where a1, a2, ……………, an are constants.
2. Scalling Theorem
If K is a constant then the Laplace transform of K f(t) is given as K times the Laplace transform of f(t).

3. Real Differentiation (Differentiation in Time Domain)
Let F(s) be the Laplace transform of f(t). Then,

Where f(0-) indicates value of f(t) at t = 0- i.e. just before the instant t = 0.The theorem can be extended for nth order derivative as,

Where f(n-1)(0-) is the value of (n-1)th derivatives of f(t) at t= 0-.

This property is most useful as it transform differential time domain equations simple algebraic equations, along with the initial conditions, if any.
4. Real Integration
If F(s) is the Laplace transform of f(t) then,

This property can be extended for multiple integrals as,

5. Differentiation by s
If F(s) is the Laplace transform of f(t) then the differentiation by s in the complex frequency domain corresponds to the multiplication by t in the time domain.

6. Complex Translation
If F(s) is the Laplace transform of f(t) then by the complex translation property,

Where F(s) is the Laplace transform of f(t).
7. Real Translation (Shifting Theorem)
This theorem is useful to obtain the Laplace transform of the shifted or delayed function of time.
If F(s) is the Laplace transform of f(t) then the Laplace transform of the function delayed by time T is,

8. Initial Value Theorem
The Laplace transform is very useful to find the initial value of the time function f(t). thus if F(s) I the Laplace transform of f(t) then,

The only restriction is that f(t) must be continuous or at the most, a step discontinuity at t = 0.

9. Final Value Theorem
Similar to the initial value, the Laplace transform is also useful to find the final value of the time function f(t). Thus if F(s) is the Laplace transform of f(t) then the final value theorem states that,

The only restriction is that the roots of the denominator polynomial of F(s) i.e. poles of F(s) have negative or zero real parts.

Table of Laplace Transforms :