Partial fraction decomposition breaks rational functions into simpler ones that we can operate on — these operations include taking the inverse Laplace transform or integrating.
For the inverse Laplace transform, we break it into a function we can convert with . For integration, we break functions into those we can integrate into or use substitution methods.
In MATLAB, we can use partfrac(exp, var), where var is optional.
General idea
We integrate something of the form: . Where the degree of the numerator is greater than or equal to the degree of the denominator, we perform long division to reduce it:
So where the degree of the denominator is greater than the numerator, we can always reduce into either linear factors or a quadratic factor or a combination. The quadratic factor will have no real roots. We can always find numbers and to make this split decomposition work:
For a quadratic term in the denominator, we use a numerator term of . If there’s an irreducible quadratic term (i.e., no real roots) in the denominator, we can complete the square to still be able to integrate or find the inverse Laplace transform.
For repeated terms in the denominator, we “step up” with the split terms’ denominator exponents.
Cover-up method
For functions where the denominator doesn’t have repeated roots, we can use a special technique called the cover-up method. We can just pick a convenient value for one of the roots and solve for the coefficient.
For example, for:
We can then assume (to solve for ), and (to solve for ). This is pretty fast! We can generalise:
Complex poles
For any polynomial in the denominator, we know that if it has a complex root, the conjugate will be another root, i.e.: