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3.6. Rational Functions 109

8. For 2 < n 5 5, write out the partial fraction decomposition of
   l/(P - 1) over Q and C. What is the decomposition over C for
   an arbitrary positive integer n?

Explorations

E.38. Principal Parts and Residues. What happens in the partial frac-

tion decomposition if the denominator q(t) of a rational function p(t)/q(t)

has repeated irreducible factors? Suppose q(t) = (t-c)kr(t) where r(c) # 0.

Then p(t)/q(t) can be written in the form

u(t)

(t - C)k

+ s(t)

*

where deg u(t) < k. To see this, note that by the Euclidean algorithm, there

are polynomials g(t) and h(t) such that

1 = g(t)(t - cy + h(t)?=(t).

(Consult Exercise 1.6.2 for the numerical case and Exploration E.20.) Di-

viding by q(t) and doing some manipulating yields the required represen-

tation.

By using the Taylor expansion of u(t) in terms of t - c, show that, for

suitable coefficients ei,

u(t)

-=~+(t”“c;:_, (t - c)k +-.+&.

This is called the principal part of the rational function at c. For t close

to c, the numerical behaviour of the rational function is approximated by

that of its principal part. The coefficient al of (t-c)-’ is called the residue

at c. Both the principal part and the residue play an important role in

the theory of functions of a complex variable; the residue can be used to

compute definite integrals which often cannot be evaluated by elementary

means.

In the special case that k = 1,

q(t) = (t - CM(t) + (t - +w

for some polynomial u(t), where q’(c) # 0. Show that the residue ofp(t)/q(t)
at c is equal to p(c)/q’(c).
More generally, if q(t) = v(t)kr(t), where v(t) is irreducible and gcd
(v(t), r(t)) = 1, we can write

p(t) - - w(t) + (^4) -
q(t) (Wk r(t)
where deg w < k deg v.