6.2 • CONTOURS AND CONTOUR INT EGRALS 213{b) The contour C that is oriented clockwise, as shown in F igure 6.14(b).
y y- I x -I x
-i - i - 1-i
(•) (b)
Figure 6.14
- Recall Ci (a) is the circle of radius r cent ered at a, oriented counterclockwise.
(a) Evaluate fct(o) z dz.(b) Evaluate fctco) z dz.
(~) Evaluate fc;(o) ~ dz. (The minus sign means clockwise orientation.)( d) Evaluate Jc; (O) ~ dz.
(e) Evaluate fc (z + 1) dz, where C is Ct (0) in the first quadrant.
(f) Evaluate fc (x^2 - iy^2 ) dz, where C is the upper half of ct (0).
(g) Evaluate fc lz - 112 dz, where C is the u pper half of ct (0).
- Let f be a continuous function on the circle {z: lz -zol = R }. Show that
fck(zo) f (z) dz= i R J~" f (zo + Rei^9 ) e'^9 d8. - Use the results of Exercise 8 to evaluate
(a) fck(•o) .!.o dz.
(b) fct;(•o) <--~ol" dz, where n f: 1 is a n integer.- Use the techniques of Example 6.11 to show t hat
(a) IIc .l--1 dzl $ i· where c is the first quadrant portion of ct (0).
(b) lfck(O) Lo!Jzldzl $ 21r ( y'(tnR/+#2).
- Evaluate fc z^2 dz, where C is the line segment from 1 to 1 + i.
12. Evaluate fc lz^21 dz, where C is given by C: z(t) = t + it^2 , for 0 $ t $ 1.
13. Evaluate f c exp z dz, where C is t he straight-line segment joining 1 to 1 + i1r.
14. Evaluate fc z exp z dz, where C is the square with vertices 0, 1, 1 + i, and i taken
with the counterclockwise orientation.
15. Evaluate f c exp z dz, where C is the straight-li ne segment joining 0 to 1 + i.