partial fractions of expressions
Let be a fractional expression, i.e., a quotient of the polynomials and such that is not divisible by . Let’s restrict to the case that the coefficients are real or complex numbers.
If the distinct complex zeros of the denominator are with the multiplicities (), and the numerator has not common zeros, then can be decomposed uniquely as the sum
where is a polynomial and the ’s are certain complex numbers.
Let us now take the special case that all coefficients of and are real. Then the (i.e. non-real) zeros of are pairwise complex conjugates, with same multiplicities, and the corresponding linear factors (http://planetmath.org/Product) of may be pairwise multiplied to quadratic polynomials of the form with real ’s and ’s and . Hence the above decomposition leads to the unique decomposition of the form
where is the number of the distinct real zeros and the number of the distinct zeros of the denominator of the fractional expression . The coefficients , and are uniquely determined real numbers.
Cf. the partial fractions of fractional numbers.
Example.
Title | partial fractions of expressions |
Canonical name | PartialFractionsOfExpressions |
Date of creation | 2013-03-22 14:20:27 |
Last modified on | 2013-03-22 14:20:27 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 29 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 26C15 |
Synonym | partial fractions |
Related topic | ALectureOnThePartialFractionDecompositionMethod |
Related topic | PartialFractionsForPolynomials |
Related topic | ConjugatedRootsOfEquation2 |
Related topic | MixedFraction |
Defines | fractional expression |