Integration 507
9. (a) If f is an entire function such that Ref(z) ~ M (M :'.:'.: 0) for all z,
show that f is a constant function.
(b) If f is an entire function such that Im f ( z) ~ 0 for all z, show that
f is a constant function.
10. Apply Liouville's theorem to show that if f is an entire function sat-
isfying lf(z)I ~ ex for all z = x + iy, then f(z) = aez, where a is a
complex constant such that lal ~ 1.- Let P(z) = aozn + aizn-l +···+an, ao -:/= 0, n :'.:'.: 1. Show that given
€ > 0 there exists R > 0 such that
(1 - €)laolrn ~ IP(z)I ~ (1 + €)laolrn
/whenever lzl = r > R. r
12. If P( z) is a polynomial of degree n :'.:'.: 1, and C is a simple closed contour
whose interior contains the zeros of the polynomial, prove that1 J P'(z)
27fi P(z) dz= n
c+- Show that the fundamental theorem of algebra is equivalent to the
property that every nonconstant complex polynomial function defines
an open mapping. (R. L. Thompson [37])
14. The function f(z) = (tanz)/z is analytic in A= {z: lzl < 1.5} except
at z = 0 where it is not defined, yet locally bounded. How f(O) is to be
defined so that the function thus extended will become analytic in A?15. Suppose that f and g are analytic in a simply connected region R, that
g^1 (z) -:/= 0 and g is one-to-one in A. If C is a simple closed contour with
graph in R and z (j. C*, show thatDc(z)f(z) = g'(z.) J J(() d(
27fi g(()-g(z)
c16. Show that F(z) = J: e-zt dt is analytic in Rez > 0, and compute F'(z).
17. If f is analytic in a region containing the unit disk lzl ~ 1, show that
f(reio) = ~ 1211' f(ei.P.) d'ljJ
27f 0 1 - re•<^8 -.P)for 0 ~ r < 1.
- Show that
----------= 27f
1211' (R2 -r2) d'ljJ