1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

212 CHAPTER 6 • COMPLEX INTEGRATION

(b) C = c, + G2 + Gs, as indicated in Figure 6.12.

I • x

3 - 3 -2 -I

Figure 6.11 Figure 6.12

2. Sket ch the following curves.

(a) z(t) = t^2 - 1 +i(t+4), for 1$t$3.

(b) z(t)=sint+icos2t,for-! $t$ ~·

(c) z(t) = 5cost-i3sint, for!$ t $ 2ir.

y

2 3

3. Consider the integral f c z^2 dz, where C is the positively oriented upper semicircle
   of radius 1, centered at 0.

(a) Give a Riemann sum approximation for t he integral by selecting n = 4

. ill i ('lk- 1 )"
and the pomts Zit = e • (k = 0,. .. , 4) and Ck = e • (k = 1 ,. .. , 4).
(b) Compute t he integral exactly by selecting a parametrization for C and
applying T heorem 6.1.
4. Show t hat the integral of Example 6.7 simplifies t o exp (2 +ii) - 1.
5. Evaluate f c x dz from - 4 to 4 along the following contours, as shown in Figures
6.13(a) and 6.13(b).

(a) The polygonal pat h C with vertices -4, -4 + 4i , 4 + 4i, and 4.

(b) The contour C that is the upper half of t he circle lzl = 4, oriented

clockwise.

y y

• 4+4i 4i 4 +4i 4i
  c 2i

x

-4 -2 2 4 -4 - 2 2 4

(a) (b)

Figure 6.13

6. Evaluate f 0 y dz for - i to i along the following contours, as shown in Figures

6.14(a) and 6.14(b).

(a) The polygonal path C with vertices -i, - 1 -i, -1, and i.