1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

202 CHAPTER 6 • COMPLEX INTEGRATION

y

2

Figure 6.6 Partition and evaluation points for the Riemann sum S (P 8 ).

I Definition 6.2: Complex Integral

Let C be a contour. Then

provided the limit exists in the sense previously discussed.

Note that in Definition 6.2, the value of the integral depends on the contour.

In Section 6.3 the Cauchy-Goursat theorem will establish the remarkable fact

that, if f is analytic, then fc f (z)dz is independent of the contour.

• EXAMPLE 6.6 Use a Riemann sum to get an approximation for the integral

fc exp z dt, where C is the line segment j oining the point A= 0 to B = 2 + i7.

Solution Set n = 8 in Equation (6-12) and form the partition P 8 : Zk =

~ + i;;, for k = 0, 1, 2, ... , 8. For this situation, we have a uniform increment

Azk = ~ + i; 2. For convenience we select ck= zo-~+·• =^2 k8l + i " <^2 ; 4 -l), for

k = 1, 2,... , 8. Figure 6.6 shows the points {zk} and {ck}·

One possible Riemann sum, then, is

8

8

[2k- 1 7r(2k- 1)](1 11")

S(Ps)={;f(ck)Azk={;exp -8-+i 64 4+i32.

By rounding the terms in this Riemann sum to two decimal digits, we obtain an

approximation for the integral:

S (Ps) ~ (0.28 + 0.13i) + (0.33 + 0.19i) + (0.41+0.29i) + (0.49 + 0.42i)

+ (0.57 + 0.6i) + (0.65 + 0.84i) + (0.72 + l.16i) + (0.78 + l.57i)

~ 4. 23 + 5.20i.