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Complex Analysis Homework 11 Solutions, Assignments of Health sciences

Solutions to homework 11 in the complex analysis course, m381d 59180. Topics covered include entire functions, zeros, and the characterization of the genus. Questions include proving the existence of entire functions f(z) and g(z) such that f(z)ϕ(z) + g(z)ψ(z) = 1, analyzing the behavior of sin(π(z + α)), and determining the genus of cos √z.

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Pre 2010

Uploaded on 08/31/2009

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Homework 11 04–09–08
Complex Analysis, M381D 59180
0. Optional. (Prelim Spring 2005) Suppose that F(z) and G(z) are entire functions
having no common zeros. Prove that there exist entire functions ϕ(z) and ψ(z) such
that
F(z)ϕ(z) + G(z)ψ(z) = 1
for all zC.
1. Assume that fis entire analytic and has nzeros (counting multiplicities) in a disk
|z|< r. Show that for every R > r there exists a complex number zof modulus R,
such that
|f(z)| |f(0)|R
r1n
.
2. (Ahlfors 5.2.3.2) Prove that
sinπ(z+α)= sin(πα)eπz cot(π α)
Y
n=−∞ 1 + z
n+αez/(n+α)
whenever αis not an integer. Hint: Denote the factor in front of the canonical product
by g(z) and determine g0(z)/g(z).
3. (Ahlfors 5.2.4.1) The characterization of the genus given in the first paragraph of
Sec. 3.2 is not literally the same as the definition in Sec. 2.3. Supply the reasoning
necessary to see that the conditions are equivalent.
4. (Ahlfors 5.2.3.3) What is the genus of cos z?
5. (Ahlfors 5.2.3.4) If f(z) is of genus h, how large and how small can the genus of f(z2)
be?
6. (Conway VII.5.13) Find a non-constant entire function f, such that f(m+in) = 0 for
all possible integers m, n. Find the most elementary solution possible.
7. (Prelim Jan 2001) Suppose that f(z) is an entire function such that |f(z)| BeA|z|,
zC, for some positive constants Aand B. Let ω1, ω2, . . . be the zeros of flisted
with appropriate multiplicity. Prove that
X
n=1
(1 + |ωn|)α<
for all α > 1.
1

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Homework 11 04–09–

Complex Analysis, M381D 59180

0. Optional. (Prelim Spring 2005) Suppose that F (z) and G(z) are entire functions

having no common zeros. Prove that there exist entire functions ϕ(z) and ψ(z) such that F (z)ϕ(z) + G(z)ψ(z) = 1

for all z ∈ C.

1. Assume that f is entire analytic and has n zeros (counting multiplicities) in a disk

|z| < r. Show that for every R > r there exists a complex number z of modulus R, such that |f (z)| ≥ |f (0)|

R

r

)n .

2. (∼ Ahlfors 5.2.3.2) Prove that

sin

π(z + α)

= sin(πα)eπz^ cot(πα)

∏^ ∞

n=−∞

z n + α

e−z/(n+α)

whenever α is not an integer. Hint: Denote the factor in front of the canonical product by g(z) and determine g′(z)/g(z).

3. (Ahlfors 5.2.4.1) The characterization of the genus given in the first paragraph of

Sec. 3.2 is not literally the same as the definition in Sec. 2.3. Supply the reasoning necessary to see that the conditions are equivalent.

4. (Ahlfors 5.2.3.3) What is the genus of cos

z?

5. (Ahlfors 5.2.3.4) If f (z) is of genus h, how large and how small can the genus of f (z^2 )

be?

6. (Conway VII.5.13) Find a non-constant entire function f , such that f (m + in) = 0 for

all possible integers m, n. Find the most elementary solution possible.

7. (Prelim Jan 2001) Suppose that f (z) is an entire function such that |f (z)| ≤ BeA|z|,

z ∈ C, for some positive constants A and B. Let ω 1 , ω 2 ,... be the zeros of f listed with appropriate multiplicity. Prove that

∑^ ∞

n=

(1 + |ωn|)−α^ < ∞

for all α > 1.