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MATH
[1251]

Theorems from Rational functions

All theorems from Rational functions.

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Partial fractions decomposition

For a fixed denominator after cancellation and factorisation, every proper rational function ff has a unique partial-fraction expansion with terms of the form

A(x−a)k,Bx+C(x2+bx+c)k,\frac{A}{(x-a)^k},\qquad \frac{Bx+C}{(x^2+bx+c)^k},

where the quadratic x2+bx+cx^2+bx+c is irreducible over the real numbers (i.e. b2−4c<0b^2-4c<0).

The integers kk run up to the multiplicity of each factor. Uniqueness means the linear equations for the undetermined coefficients have exactly one solution. The real irreducible quadratics arise by pairing a non-real complex root with its conjugate, as in real linear and quadratic factorisation.

  1. [1]:Proper rational function
  2. [2]:Partial fraction terms
  3. [3]:Integrate a rational function
  4. [4]:Distinct linear factors
  5. [5]:Repeated linear factors
  6. 4 more

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Partial fractions decomposition