# Basic Theorems of Laplace Transform

1. Linearity Theorem If Laplace transform of the functions f1(t) and f2(t) are F1(s) and F2(s) respectively, then Laplace transform of the functions [K1 f1(t) + K2 f2(t)] will be [K1 F1(s) + K2 F2(s)].

[K1 f1(t) + K2 f2(t)] = [K1 F1(s) + K2 F2(s)]

where, K1 and K2 are constants.
2. Scaling Theorem If Laplace transform of f (t) is F(s), then

[f(Kt)] = , where K is a constant and K > 0.
3. Time Differentiation Theorem If Laplace transform of f (t) is F(s), then,

4. Frequency Differentiation Theorem If Laplace transform of f (t) is F(s), then,

5. Time Integration Theorem If Laplace transform of f (t) is F(s), then,

In general, for nth order integration,

6. Shifting Theorem The shifting may be done with respect to time or frequency.
1. Time Shifting Theorem

If Laplace transform of f (t) is F (s), then

2. Frequency Shifting Theorem

If Laplace transform of f(t) is F(s), then

7. Initial Value Theorem If the Laplace Transform of f (t) is F (s) and the first derivative of f (t) is Laplace transformable, then, the initial value of f (t) is,

8. Final Value Theorem If a function f (t) and its derivatives are Laplace transformable, then the final value of f (t) is,