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partial fractions of expressions∗
pahio†
2013-03-21 17:22:38
P (z)
Let R(z) = Q(z)
be a fractional expression, i.e., a quotient of the polynomials 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 numbers.
If the distinct complex zeros of the denominator are b1 , b2 , . . . , bt with the
multiplicities τ1 , τ2 , . . . , τt (t ≥ 1), and the numerator has not common zeros,
then R(z) can be decomposed uniquely as the sum
R(z) = H(z) +
t X
Ajτj
Aj1
Aj2
+
+
.
.
.
+
,
z − bj
(z − bj )2
(z − bj )τj
j=1
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 imaginary (i.e. non-real) zeros of Q(z) are pairwise complex
conjugates, with same multiplicities, and the corresponding linear factors of
Q(z) may be pairwise multiplied to quadratic polynomials of the form z 2+pz+q
with real p’s and q’s and p2 < 4q. Hence the above decomposition leads to the
unique decomposition of the form
m X
Ai1
Aiµi
Ai2
+
.
.
.
+
+
x − bi
(x − bi )2
(x − bi )µi
i=1
n X
Bjνj x + Cjνj
Bj1 x + Cj1
Bj2 x + Cj2
+
+ 2
+ ... + 2
,
x2 + pj x + qj
(x + pj x + qj )2
(x + pj x + qj )νj
j=1
R(x) = H(x) +
where m is the number of the distinct real zeros and 2n the number of the
distinct imaginary zeros of the denominator Q(x) of the fractional expression
P (x)
. The coefficients Aik , Bjk and Cjk are uniquely determined real
R(x) = Q(x)
numbers.
Cf. the partial fractions of fractional numbers.
∗ hPartialFractionsOfExpressionsi
created: h2013-03-21i by: hpahioi version: h35812i
Privacy setting: h1i hDefinitioni h26C15i
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1
Example.
−x5 +6x4 −7x3 +15x2 −4x+3
1
x
3
2x−1
= −
+ 2
+
+ 2
3
2
2
3
(x−1) (x +1)
x−1 (x−1)
x +1 (x +1)2
2