1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

194 CHAPTER 6 • COMPLEX INTEGRATION

We generally evaluate integrals of this type by finding the antiderivatives of

u and v and evaluating the d efinite integrals on the right side of Equation (6-1).

That is, if U' (t) = u(t), and V' (t) = v(t), for a~ t ~ b, we have

rb

1

t-b

lo. j(t)dt=[U(t)+iV(t)J t=a =U(b)-U(a)+i[V(b)-V(a)].

• EXAMPLE 6.1 Show that

r1 ,3 - 5

10

(t-i)dt=

4

.

(6-2)

Solu tion We write t he integrand in terms of its real and imaginary parts, i.e.,

f (t) = (t - i)^3 = t^3 - 3t+i (-3t^2 + 1). Here, u (t) = t^3 -3t and v (t) = - 3t^2 + 1.

The integrals of u and v are

fl ( 3 ) - 5 r1

lo t - 3t dt = 4 and lo (-3t

2

+ 1) dt = O.

Hence, by Definition (6- 1 ),

(t-i)^3 dt= u(t)dt+i v(t)dt=-.
1

1 11 11 -5

0 0 0 4

• EXAMPLE 6.2 Show that

lo ['i exp(t+it)dt=1 ( • ) i ( • )

2

e•- 1 +

2

e•+l.

Solution We use the method suggested by Definitions (6-1) and (6-2).