Rational functions - Cs.ioc.ee

Rational Expansion Theorem Jaan Penjam
7 Complex numbers vs real numbers
Everything above is valid for the polynomials over complex fields, i.e. all variables and coefficients are
complex numbers. The methods described work also for real numbers, but only if the polynomial Q(z)
has ℓ real roots, where degQ(z) = ℓ. The assumption that a polynomial has as many roots as its degree is
guaranteed for complex numbers by the Fundamental Theorem of Algebra. This is the reason why you
are strongly suggested to apply complex analysis in this context.
However, it is possible to keep yourself to real numbers as well, but you need to consider possibly
much more complex calculations. For example, to decompose a rational function into partial fractions
within real numbers, you have to consider, that a real polynomial Q(x) can have two types of irreducible
factors:
• x − ρ, where ρ is a real root of Q(x);
• x 2 + px + q, where p and q are real numbers such that p 2 < 4q.
Factors of both types can appear multiple times in the polynomial. Recall the techniques of such decomposition
in notes of your undergraduate course of mathematical analysis, for example pages 111–118 in
http://staff.ttu.ee/~janno/ma1chem.pdf.
To use method of Decomposition into Partial Fractions parts of the rational function R(x) = P(x)/Q(x)
are of the form
•
A1 A2
x−ρ + (x−ρ) 2 + ··· + Ak
(x−ρ) k , for a root ρ ∈ R with multiplicity k;
• B1+C1x
x2 B2+C2x
+ +px+q (x2 +px+q) 2 + ··· + Bk+Ckx
(x2 +px+q) k for a factor x2 + px + q with multiplicity k.
All constants Ai,Bi,Ci should be found by solving the system of linear equations similarly to case of
complex numbers (see Example 6.1).
After decomposition, parts of the form A1
x−ρ can be transformed into the poewer series similarly to
the formula 2:
1
n + k − 1 xk =
(x − ρ) k k − 1 ρk 1
(−ρ) k (1 − 1
ρ x)k = (−1)k · 1
ρ k ∑ n0
Unfortunately, in general, the parts of the form B+Cx
x 2 +px+q
does not have a simple transformation into
power series. For particular numbers p and q the assistance of an one-line Taylor Series Calculator
may be helpful. See for example, the Wolfram|Alpha Widgets on http://www.wolframalpha.com/
widgets/view.jsp?id=f9476968629e1163bd4a3ba839d60925. Some first terms of decomposition
of the fraction 1/(x 2 + px + q) looks like:
1
x2 1 px
= −
+ px + q q q2 + (p2 − q)x2 q3 − (p3 − 2pq)x3 q4 mmmmmmmmmmmmmmmmmmmjj − (p5 − 4p 3 q + 3pq 2 )x 5
7
q 6
+ (p4 − 3p2q + q2 )x4 q5 −
+ (p6 − 5p 4 q + 6p 2 q 2 − q 3 )x 6
q 7
− ···