The laplace transform transforms a function from the time domain to the laplace domain. For a continuous function . with , the laplace transform is defined as
The notation used is
Where is the function in the laplace domain. Tables of laplace transforms for common functions are given in the formula book, so there is no need to work out most transforms manually.
Transforms are linearly independent in the same way integrals are:
For example, find the laplace transform of :
Transforms also have an inverse:
For example, find from
Sometimes, partial fractions and/or completing the square is required to get the equation into a form recognisable from the table.
Laplace transforms exist of derivatives:
This can be used to solved differential equations, by laplace transforming the differential equation to make an algebraic one, then inverse laplace transforming the result back.
Need to use partial fractions to inverse transform
Taking inverse laplace transforms using table: