Laplace Transform

Let be defined for all . Its Laplace transform is:

where , provided that integral exists.

If additionally for , then (Fourier Transform)

Examples

• ,
• ,
• ,
• ,

Properties

• Linearity
• Translation
• Scaling
• Shifting
• Transform of derivatives
• Derivative of transform
• Asymptotic limits: if exists:
• Convolution: , where is Convolution of these functions.

Inverse

We can invert the Laplace transform using the Bromwich inversion formula:

where is chosen to be greater than the real part of all singular points of .

A more handy formula is:
If has only finitely many isolated singularities and then for , , while for :

Differential equations

We can apply Laplace transform to differential equations to solve them:

Applying Laplace transform:

Now apply boundary conditions and solve this for . Then invert to find .