198 CHAPTER 6 • COMPLEX INTEGRATION- Show that Jo"" e-•'dt = ~ provided Re (z) > O.
- Establish the following identities.
(a) Identity (6-3}.
(b} Identity (6-4}.
(c) Identity (6-6}.
(d} Identity (6-7).- Let f (t) = u (t) + iv (t}, where 'ti and v are differentiable. Show that
1: J(t}J'Ct}dt= !ff(b)J^2 -HJ<a)]^2. - Use integration by parts to verify that if 0 lf e'sintdt= ~ (e~ + 1).
6.2 Contours and Contour Integrals
In Section 6.1 we showed how to evaluate integrals of the form J: f (t) dt , where
f was complex-valued and [a, b] was an interval on the real axis (so that twas
real, with t E [a, b)). In this section, we define and evaluate integrals of the form
fc J (z) dz, where f is complex-valued and C is a contour in the plane (so that
z is complex, with z EC). Our main result is Theorem 6.1, which shows how to
transform t he latter type of integral into the kind we investigated in Section 6. 1.
We use concepts first introduced in Section 1.6. Recall that to represent a
curve C in the plane we use the parametric notationC: z(t) = x(t) +iy(t), for a ::; t ::;· b, {6-10)where x (t) and y (t) are continuous functions. We now place a few more restric-
t ions on the type of curve to be described. The following discussion leads to the
concept of a contour, which is a type of curve that is adequate for the study of
integration.
Recall that C is simple if it does not cross itself, which means that z (ti) f=
z (t2) whenever t1 f= t2, except possibly when t 1 = a and t2 = b. A curve C
with the property z (b) = z (a) is a closed curve. If z (b) = z (a) is the only
point of intersection, then we say that C is a simple closed curve. As the
parameter t increases from the value a to t he value b, the point z (t) starts at
the initial point z ( a), moves along the curve C, and ends up at the terminal
point z (b). If C is simple, then z (t) moves continuously from z (a) to z (b) as t
increases and the curve is given an orie ntation, which we indicate by drawing
arrows along the curve. Figure 6.1 illustrates how the terms simple and closed
describe a curve.
The complex-valued function z (t) = x (t)+iy (t) is said to be differentiable
on [a, b] if both x (t) and y (t) are differentiable for a::; t::; b. Here we require the