486 Chapter 7
14. Let C be a simple closed contour and suppose that f is analytic on
C U Ext C, and that limz-+oo f ( z) = L. Prove that if a E Ext C*,
we have~ J f(z)dz = f(a)-L
27TZ Z - ac-
and if a E Int C*, then_l J f(z)dz = L
27Ti z - a
c+
In the first formula c- denotes the contour C described once in
the negative direction (this formula is"called Cauchy's integral for an
unbounded region).15. Suppose that C and f satisfy the conditions stated in problem 14 and,
in addition, assume that 0 E Int C*. Show that_l j af(z) dz={O
27Ti az-z^2 f(a)
c+- Prove formula (7.17-10).
if a E Int C*
if a E Ext C*17. (a) Show that a function f E C^1 (A), A open, can be approximated at
a point z E A by a function analytic at z with errorwhere ( = e + i17 and j) = { (: IC - zl :::; p} c A.
(b) Let ti= ma1C lfi;(()I be called the deviation off from analyticity
(ED
in fJ. Show that IEI :::; 2tip.18. Let f be analytic on and within a simple closed contour C. Suppose
that maxzEC lf(z)I = M. Prove that lf(z)I:::; M for any z E IntC*.
Hint: Let r = d(z, C*) and n a positive integer. Show that lf(zW :::;
Mn /27rr. Then take the nth root on both sides and let n --t oo.7.20 Integrals of Cauchy Type
Definition 7.11 Integrals of the form
J
f(()d(
(-z
'"'(