1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

6.2 • CONTOURS AND CONTOUR INTEGRALS 203

This result compares favorably with the precise value of the integral, which you
will soon see equals

exp ( 2 + i~) - 1 = -1 + e^2 v; + ie^2 v; ~ 4.22485 + 5.22485i.

In general, obtaining an exact value for an integral given by Definition 6.2 is

a daunting task. Fortunately, there is a beautiful theory that allows for an easy

computation of many contour integrals. Suppose that we have a parametrization

of the contour C given by the function z (t) for a ~ t ~ b. That is, C is the

range of the function z (t) over the interval [a, b], as Figure 6.7 shows.

It follows that

n n

n-lim ooL '°' f (ck) f:::,.zk = n-lim ooL '°' f (ck) (zk - Zk-1)

k=l k=I

n

= J~~ L f (z (rk)) [z (tk) - z (tk- 1)],

k = l

where 7'Jc and tk are the points contained in the interval [a, b] with the property

that Ck= z(rk) and zk = z(tk), as is also shown in Figure 6.7. If for all k we

multiply the kth term in the last sum by :::::~:::, then we get

The quotient inside the last summation looks suspiciously like a derivative,

and the entire quantity looks like a. Riemann sum. Assuming no difficulties, this

y

Zn- 1 Zn= B

= z(b)

Zo=A

= z(a)

..... - ... -... ·--.. ··-- -+--------------+x

Figure 6. 7 A parametrization of the contour C by z ( t) , for a ~ t ~ b.