Preliminary Examination 1999
Complex Analysis
Do all problems.
Notation:
¡ = {x : x is a real number}
B ( a , r ) = { z ∈ £ :| z − a | < r}
D = B(0,1)
£ = { z : z is a complex number}
ann( a , r1 ,r2 ) = { z ∈ £ : r1 < | z − a | < r2}
For G ⊂ £ , let A (G ) denote the set of analytic functions on G (mapping G to £ ) and let
Har (G ) denote the set of harmonic functions on G (mapping G to ¡ ) .
∞
1.
∑a z
Suppose the power series
n= 0
n
n
converges on | z | < R where z and the an are complex
numbers. If bn ∈ £ is such that | bn | < n | a n | for all n, prove that
2
∞
∑b z
n= 0
n
n
converges for
|z|<R.
2.
Let f , g ∈A ( D ) . For 0 < r < 1, let Γ r = ∂B (0, r ) .
a)
Prove that the integral
1
2π
∫
Γr
1
z
f ( w )g ( ) dw is independent of r provided
w
w
that | z | < r < 1 and that it defines an analytic function h ( z ), | z | < 1 .
b)
3.
Prove or supply an counterexample: if f
Let f ∈A ( D ) . Suppose that | f ( z ) | ≤
of f ) such that | f ′( z ) | ≤
4.
M
(1− | z |)
3
0 and g
1
(1− | z |)
1
0, then h
0.
. Prove that there exists a M (independent
2
.
2
Let f ∈A ( D ) such that f ( D) ⊂ D, f (0) = 0. Prove for z ∈ D that
| f ( z ) + f ( −z ) | ≤ 2 | z |2 with equality if and only if f ( z ) = λ z 2 for some λ with | λ |= 1 .
5.
In which quadrant do the roots of p ( z ) = z 4 + 2 z + 1 lie?
6.
Let f ∈A ( D ) and suppose that f is not the identity map. How many fixed points can f
have?
7.
Let f be an entire function. Suppose that f has a root at z = + i and z = −i . Let
M = max| f ( z ) | . Prove that | f ( z ) | ≤
| z |= 2
∞
8.
Prove that
∏ (1 −
n =1
∞
9.
Evaluate
∫
0
M 2
| z + 1| on B (0,2) .
3
z2
) is an entire function and find its zeros, counting multiplicity.
n2
x sin x
dx .
x2 +1
10. Let u ∈Har ( £) . Show that u can be positive on all of £ only if u is constant.