Partial fraction decomposition
Partial fraction decomposition or partial fraction expansion is a method to rewrite quotients of two polynomials and , where the numerator is of lower order than , as sums of fractions with in the denominator:
Where etc. are the roots of the denominator . If all of these roots are distinct, then it is sufficient to simply posit:
The constants can either be found the hard way, by multiplying the denominators around and solving a system of equations, or the easy way by using this trick:
If is a root with multiplicity , then the sum takes the form of:
Where are found by putting the terms on a common denominator, e.g.
And then, using the linear independence of , solving a system of equations to find all .
- O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.