The Role of The Taylor Polynomial
The partial fraction decomposition of a rational function can be related to Taylor's theorem as follows. Let
be real or complex polynomials; assume that
that
and that
Define also
Then we have
if, and only if, for each the polynomial is the Taylor polynomial of of order at the point :
Taylor's theorem (in the real or complex case) then provides a proof of the existence and uniqueness of the partial fraction decomposition, and a characterization of the coefficients.
Sketch of the proof: The above partial fraction decomposition implies, for each 1 ≤ i ≤ r, a polynomial expansion
- , as
so is the Taylor polynomial of, because of the unicity of the polynomial expansion of order, and by assumption .
Conversely, if the are the Taylor polynomials, the above expansions at each hold, therefore we also have
- , as
which implies that the polynomial is divisible by
For also is divisible by, so we have in turn that is divisible by . Since we then have, and we find the partial fraction decomposition dividing by .
Read more about this topic: Partial Fraction
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