Laplace Transform

A mathematical operation that transforms a function of time , into a function of a complex variable, denoted as . Mathematically, the Laplace Transform of a function is defined as:

Here:

• represents time
• , where
• is the exponential decay factor that weights the function .

Can only be applied to casual functions. Converts a differential equation (time domain) into a linear complex algebraic equation (frequency domain). After Laplace transformation is applied, the resulting functions are capitalized by convention.

Casual function

A function of time satisfying:

Inverse Laplace transform

If , then the inverse Laplace Transform is given by:

Special functions

Unit step function

Laplace transform of the unit step function is:

Unit impulse function

Laplace transform of the unit impulse function is:

Area under the curve is .

And it has a special property:

Unit ramp function

Laplace transform of the unit ramp function is:

Properties

Suppose has a Laplace transform for the below definitions.

Linearity

and are constants.

Differentiation

Integration

Time Scaling

Frequency Scaling

Multiplication by t

Time shift

Frequency shift

Theorems

Initial Value Theorem

Final Value Theorem

Laplace transform table

Function Name	Function	Laplace Transform
Unit Impulse
Unit Step
Polynomial
Exponential
Sine Wave
Cosine Wave
Damped Sine Wave
Damped Cosine Wave
Sinh Wave
Cosh Wave
Damped Sinh Wave
Damped Cosh Wave
When
When

Note

The above table will be given in final examination. And don’t need to be memorized.