Partial Fraction - The Role of The Taylor Polynomial

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 :

A_i(x):=\sum_{k=0}^{\nu_i-1} \frac{1}{k!}\left(\frac{P}{Q_i}\right)^{(k)}(\lambda_i)\ (x-\lambda_i)^k.

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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