440 Chapter 7
(b)
J
Log z z dz 1f ( R 1/ ) h it 1/
2 :::; R Log +^2 7f , w ere 7: z = Re , -^2 7r :::; t :::;
27fe-aR
< R , where a > 0 and 7: z
1-e-
-R +it,
0 :::; t :::; 27f
( d) If 7: z = z(t), a :::; t :::; /3 is rectifiable, then
lz(/3) - z(a)I:::; J ldzl
'Y
- Suppose that the real-valued functions P(x,y) and Q(x,y) are defined
and continuous in a simply connected region R, and that the partial
derivatives Py and Qx exist and are continuous on R. If C is any simple
closed contour with graph contained in R, show that
J Pdx+Qdy = 0
G
iff Py = Qx, in which case P dx + Q dy is an exact differential.
8. Suppose that f is analytic in a simply connected region that contains
the graph of the simple closed contour C. Show that the value of the
integral
J f(z)f'(z) dz
G
is a pure imaginary number.
- Suppose that u1 and u 2 are harmonic functions in a simply con-
nected region R, and that v 1 and v 2 are the corresponding harmonic
conjugates. Prove that
j(u1 dv2 - u2 dv1) = o
G
for every closed contour C contained in R.
- Let A be the area bounded by the simple closed contour C. Show that:
(a) A = -i J x dz (b) A = -J y dz
G G
where C is described once in the positive direction.
- Compute the area bounded by the astroid
z = a( cos^3 t + i sin^3 t), a > 0, 0 :::; t :::; 27f
