|
||
|
|
||
|
We want to write a general rational function P(x)/Q(x) as a sum of integrable terms.
Suppose the degree of P(x) is p and that of Q(x) is q. If p is at least q, we can ide P by Q (by synthetic, ie, long division) to obtain a polynomial, D(x), and a remainder, R(x)/Q(x), with the degree, r,of R less than q.
The fraction R(x)/Q(x) then approaches 0 asYxYincreases in every direction in the complex plane.
Every polynomial can be factored into linear factors (if complex numbers can appear in the factors).Therefore we can write:
where mk is the multiplicity of the kth root of Q(x); the sum's of the m's being q.
The partial fraction theoem:
You can write R(x) / Q(x) as a sum over the roors q j of the terms of the form
for appropriate constants a jk.
Notice that this theorem allows us to integrate P(X) / Q(x); we need only to integrate the polynomial D(x) and the various inverse powers occuring in this sum, (assuming we can compute the a jk here.)