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单词 PartialFractionsOfExpressions
释义

partial fractions of expressions


Let  R(z)=P(z)Q(z)  be a fractional expression, i.e., a quotient of the polynomialsPlanetmathPlanetmath P(z) and Q(z) such that P(z) is not divisible by Q(z).  Let’s restrict to the case that the coefficients are real or complex numbersMathworldPlanetmathPlanetmath.

If the distinct complex zeros of the denominator are  b1,b2,,bt  with the multiplicities  τ1,τ2,,τt (t1), and the numerator has not common zeros, then R(z) can be decomposed uniquely as the sum

R(z)=H(z)+j=1t(Aj1z-bj+Aj2(z-bj)2++Ajτj(z-bj)τj),

where H(z) is a polynomial and the Ajk’s are certain complex numbers.

Let us now take the special case that all coefficients of P(z) and Q(z) are real.  Then the (i.e. non-real) zeros of Q(z) are pairwise complex conjugatesMathworldPlanetmath, with same multiplicities, and the corresponding linear factors (http://planetmath.org/Product) of Q(z) may be pairwise multiplied to quadratic polynomials of the form  z2+pz+q  with real p’s and q’s and  p2<4q.  Hence the above decomposition leads to the unique decomposition of the form

R(x)=H(x)+i=1m(Ai1x-bi+Ai2(x-bi)2++Aiμi(x-bi)μi)
+j=1n(Bj1x+Cj1x2+pjx+qj+Bj2x+Cj2(x2+pjx+qj)2++Bjνjx+Cjνj(x2+pjx+qj)νj),

where m is the number of the distinct real zeros and 2n the number of the distinct zeros of the denominator Q(x) of the fractional expression  R(x)=P(x)Q(x).  The coefficients Aik, Bjk and Cjk are uniquely determined real numbers.

Cf. the partial fractions of fractional numbers.

Example.

-x5+6x4-7x3+15x2-4x+3(x-1)3(x2+1)2=-1x-1+3(x-1)3+xx2+1+2x-1(x2+1)2
Titlepartial fractions of expressions
Canonical namePartialFractionsOfExpressions
Date of creation2013-03-22 14:20:27
Last modified on2013-03-22 14:20:27
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id29
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 26C15
Synonympartial fractions
Related topicALectureOnThePartialFractionDecompositionMethod
Related topicPartialFractionsForPolynomials
Related topicConjugatedRootsOfEquation2
Related topicMixedFraction
Definesfractional expression
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