Calculus: Analytic Geometry and Calculus, with Vectors

518 Exponential and logarithmic functions

in which it is supposed that a and b are real or complex numbers for which b tea.

For example, use of (7) gives

(8)

x6 x x4

• x1)(x - x2)2(x - xa)a (x - xl)(x - x2) (x - x2)(x - x3)3
  I X2(X - xl) - x1(x - x2) x4
  X2 - x1 (x - xl)(x - x2) (x - x2)(x - x3)3
  1 x, xl x4
  x2 - x1 Lx - x2 x -xli(x - x2)(x - x3)3
  X2 X4 xl X4
  x2 - xl (x - x,)2(x - x33) x2 - x1 (x - xl)(x - x2)(x - xa)i

Even when Q(x) and R(x) are values of functions that are not polynomials, (5),

(6), and (7) enable us to express quotients as sums of simpler quotients. It is

quite easy to see that the partial fraction theorem can be proved by repeated

applications of (5), (6), and (7). Moreover, it is sometimes better to use (5)

and (6) and (7) than to use other methods for obtaining partial fraction expan-

sions. Textbooks in algebra show that if the coefficients ao, al, ., a in

(1) are real, then the zeros of the polynomial "come in conjugate pairs." This

means that if p + iq is a zero for which p and q are real and q 0 0, then p - iq

is another zero. On account of this fact the right side of (4) can, when P is real,

be represented as a sum of terms of the forms

(9)

B D Gx

(x - C)m' (x2 + Ex + F)m' (x2 + .Ex + F)m'

where m is a positive integer and the coefficients in the denominators are all

real. The quadratic denominators are all real. The quadratic denominators

arise because if Hl and H2 are constants, then there exist other constants Ha and

H4 for which

(10)

Hl Hz _ Ha + H4x

x-(p+iq) x- (p - iq) x2-2px+pt +g2

In most practical applications of this material, the numberspi, p2, ... , pk

defined above are all 1 and (3) and (4) reduce to the much simpler formulas

(11) P(x) = ao(x - xl)(x - x2) ... (x - x,,)

(12) Q(x) Al A2 11,E

P(x) x-xlx-x2+ ... +x+ - x

Except when P(x) has only real zeros, we mustuse complex numbers to achieve

simplicity.

9.5 Integration by parts In this book, and elsewhere in the scientific

world, we frequently encounter situations where effective use can be

made of the formula for integration byparts. Assuming that u and v