The Laplace transform of a function is defined by the improper integral
where s is a complex number
The purpose of the Laplace transform is to take a real function of a variable (often time, sometimes is used for other properties) and transform it into a complex function of , often representing frequency. One of the most common applications of the Laplace transform is in solving differential equations, as it converts an equation of varying degrees of differentiation into a polynomial of varying degrees of power.
Inverse Laplace transform
Given a , finding a function of which satisfies is called taking the inverse Laplace transform. This process is often far more difficult than finding a Laplace transform.
For ordinary differential equations, it is usually sufficient to find an inverse Laplace transform by finding a similar Laplace transform (preferably one with a similar denominator) using a table. Often, this requires some algebraic manipulation. For example:
By the complex inversion formula, the inverse Laplace transform is equal to the contour integral
As the Laplace transform of any function goes to zero as r approaches infinity, and all residues can found to the left of some constant, by the residue theorem this is simply equal to
Using Laplace transforms to solve differential equations
Since the Laplace transform of any derivative will include , one can solve for this and take the inverse Laplace transform to find the original function . For example, given the very simple second order differential equation
with initial conditions
Applying the Laplace transform to the differential equation yields
Now by taking the inverse Laplace transform, we get
Which does indeed satisfy the conditions set by the initial problem. This is a fairly elementary example which can easily solved by other methods, but the Laplace transform can be used in many situations where this is not the case, such as
We can expand this by partial fraction expansion to get
Now by taking the inverse Laplace transform, we can find: