**Laplace Theorem:**

The Laplace theorem is given by

**Differentiation Theorem****Integration Theorem****Differentiation of Transforms****Integration of transforms****First Shifting Theorem****Second Shifting Theorem****Initial Value Theorem****Final Value Theorem**

**(a) Differentiation Theorem:**

If a function f(t) is piecewise continuous, then the Laplace transform of its derivative d/dt [f(t)] is given by

**(b)Integration Theorem: **

If a function f(t) is continuous, then the Laplace transform of its integral ∫ f(t)dt is given by

**(c) Differentiation of Transforms:**

If the Laplace transform of the function f(t) exists, then the derivative of the corresponding transform with respect to s in the frequency domain is equal to its multiplication by t in the time domain.

**(d) Integration of transforms:**

If the Laplace transform of the function f(t) exists, then the integral of corresponding transform with respect to s in the complex frequency domain is equal to its division by t in the time domain.

**(e) First Shifting Theorem: **

If the function f(t) has the transform F(s), then the Laplace transform of

e^{-at} f(t) is F(s + a)

**(f) Second Shifting Theorem:**

If the function f(t) has the transform F(s), then the Laplace transform of

f(t – a)u (t – a) is e^{-as} F(s)

**(g) Initial Value Theorem:**

If the function f(t) and its derivative f′(t) are Laplace transformable then

**(h) Final Value Theorem:**

The final Laplace Theorem is Final Value Theorem. If f(t) and f′(t) are Laplace transformable, then