Calculus: Analytic Geometry and Calculus, with Vectors

9.4 Partial fractions 517

11 Partial fraction expansions have their principal applications in electrical
engineering and elsewhere where complex numbers (numbers of the form a + ib,
where a and b are real and i is the imaginary unit for which i2 = -1) are system-
atically used. When complex numbers are used, it is not necessary to bother
with factors (like x2 + 1 and x2 + x + 1) that cannot be factored into real
linear factors. Theory and applications can be based upon a version of the
fundamental theorem of algebra which says that if

(1) P(x) = aoxn + alxn-1 + ... +an-1x + an,

where n is a positive integer and ao ; 0, then there exist k different numbers

x3, x2,. , xk (which are not necessarily real) and k exponents p1, p2i

pk (which are positive integers) such that

(2) p1+p2+ +pk=n

and

(3) P(x) = ao(x - x1)"(x - x2)P2 ... (x - xk)Pk.

This theorem is a simple corollary of general theorems that appear in the theory

of functions of a complex variable. Less sophisticated but more complicated

proofs can be given. The numbers x1, x2, , xk are said to be zerosof the

polynomial of multiplicities p1, p2,- ' -, P. It can be proved that if Q is a

polynomial of degree less than that of P, then Q(x)/P(x) is representable in the

form

( Q(x)_ Al., 41,2 41,3 41,P1

) P(x) (x - xl) + (x - x1)2 + (x - x1)3+ +(x- xl)PI

42,1 42.2 42,3 42,Pi

(x - x2) (x - x2)2 + (x - x2)3+

+(x - x2)P!

+

4,1

+

4.2

+

4,3

+ +

4k.k

(x - xk) (x - xk)2 (x - xk)3 (x- ,xk)Pk

where the pi constants 141,1, 41.2,', - ,41,P, "go with the powers of the factor

(x - xl)," the P2 constants 42,1, 42,2, ... , 42.P, "go with the powers of the

factor (x - x2)," etcetera. Supply and demand do not generate heavy traffic

in proofs of this partial fraction theorem, but we can pause to look at the three

potent identities

(5)

(6)

and

(7)

x R(x) (x - a) + a R(x)

x - a S(x) x - a S(x)

1 R(x)^1 (x - a) - (x - b) R(x)

(x - a) (x - b) S(x) b - a (x - a)(x - b) S(x)

x R(x)^1 b(x - a) - a(x - b) R(x)

(x-a)(x-b)S(x) -h-a (x-a)(x-b) S(x)