506 Chapter^7
Now, by equating real parts in (7.28-17), we obtain
( )
= I_ 1+00 yu(t, 0) dt = I_ 1+00 yu(t, 0) dt
ux,y 7r -oo 1t-zj2 7r -oo (t-x)Z+yzand by equating imaginary parts in (7.28-18), we have
v(x ) = I_ 1+00 (x -t)u(t, 0) dt = I_ 1+00 (x -t)u(t, 0) dt
,y 7r -oo 1t-zl^2 7r _ 00 (t-x)^2 +y^2Exercises 7 .5
1. Prove the following version of Morera's theorem: If f is continuous on
the simply connected region G, and ifJ f(z)dz = 0
Tfor every triangle T such that T U Int T CG, then f is analytic in G.
2. Let f E C^1 (G), Gopen, and suppose that for every z 0 E G we have
jJ(z)dz=O
'Yr
on a sequence of circles 'Yr centered at z 0 with radii r ~ 0. Apply the
complex form of Green's theorem to show that fz(zo) = 0, and hencethat f is analytic in G.
- Prove that a nonconstant entire function comes arbitrarily close to
every complex number. - Let f be an entire function and suppose that lf(z)I :::; Mr°' for lzl =
r ~ r 0 , where M > 0 and 0 < a < 1. Prove that f reduces to a
constant function.
This property and the following exercise show that we need not
assume that lf(z)I is bounded in Liouville theorem, only that its growth
is sufficiently slow.- If f is an entire function and lf(z)I < 1 + jzj^112 for all z E C, prove
that f is a constant function.
6. Prove that an entire function f is a polynomial of degree not exceeding
n i:ff lz-n f(z)I < M for some M > 0 and all z with jzj sufficiently large.
7. If f is an entire function and lf(z )I ~ 1 for all z, prove that f is a
constant function.- Let f be an entire function and suppose that f(z) = f(z + c1) =
f(z + ic 2 ), where c 1 and c 2 are nonzero real constants. Show that f
is a constant function.