484 Chapter^7
3. Suppose that f is analytic on and within the simple closed contour C,
and .z f/. C*. Evaluate the following.(a) .~ J cos((-z) f(() d(
27ri ( - z
c
b ~1 J f(()d(
() 27ri ((-a)((-b) (a f= b, a f/. C*, b f/. C*)
c+- Evaluate fc+ {[J(z) - f(a)]/(z - a)} dz, where f is analytic on and
within the simple closed contour C, and a f/. C.
5. (a) If f is analytic on and within the circle C: z-z 0 = reit, 0::::; t::::; 271",
show that
1 [2tr
f(zo) =
2
71" lo f(zo +re it) dt
i.e., the value that f assumes at the center of the circle is the
integral mean of the values assumed by f on the circumference
(Gauss, 1839). ,
(b) Deduce a corresponding mean value theorem for the harmonicfunction u = Ref.
6. Let f and g be analytic on and within the simple closed contours C1
and 02 , respectively, with Ci c Ext C2 and C2 c Ext Ci. Show that
the expression~ [ J J(()d( + J g(()d(l
2n ( - z ( - z
ct ct
represents either f(z) or g(z), depending on whether z E Int Ci or
z E Int C;_- Evaluate fc ldzl/lz -al^2 , where C: z = reit, 0::::; t::::; 271", and !al f= r.
- Let C: z = laleit, a f= 0, 0::::; t::::; 271". Evaluate the following.
(a) (PV) J !:!. (b) (PV) J z dz
z -a z^2 + a^2
c c - Let C: z = ei^8 , 0 ::::; () ::::; 271". Show that
J ek:n dz= 271"i
c
where k is a real constant and n a positive integer. Then obtain the
formula
[2tr
lo e" cos n^9 cos( k sin n8)d() = 271"