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MAT334 Summer 2023

Midterm 2

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Problem 1. (10 pts)

1. (4 pts) Let R1,R2 > 0 and R = Rei, ∈ [0,2] the circle around 0 with radius R. Give

a explicit homotopy between R1 and R2 and show it is a homotopy.

2. (1 pt) Show explicitly the homotopy between the circle of radius 1 centered at the origin

and the point z = i.

3. (5 pts) Consider a curve ∶ [a, b]→ C, continuously di?erentiable on (a, b) and continu-

ous on [a, b], define the conjugate curve ? ∶ [a, b]→ C by ?(t) = (t) show that for every

continuous function f , ?

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Problem 2. (10 pts)

1. (5 pts) Let f1, f2 ∶ C→ C be two holomorphic functions on all of C, assume that f2(z) ≠ 0

for every z ∈ C. Show that if f1(z) ≤ f2(z) then there exists a complex number w such

that f1(z) = w ? f2(z) for all z ∈ C.

2. (2.5 pts) Let p > 1, assume that for every k ∈ N, fk ∶ A → C is a function where A is a

region, assume kpfk(z) ≤ 2023 for every z ∈ A. Show that

∞

k=1 fk(z) converges absolutely for every z ∈ A.

3. (2.5 pts) Show that every polynomial p ∶ C→ C of degree at least 1 is surjective.

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Problem 3. (10 pts)

1. (6 pts) Assume that f ∶ C → C is di?erentiable at every point and there exists K, a

compact set such that for every z ∈ CK there exists wz ∈K such that f(z) ≤ f(wz).

Show that f ′(z) = 0 for every z ∈ C.

2. (4 pts) Let A = {z ∈ C ∶ R(z) > 1}, let B = {z ∈ C ∶ z ? z0 ≤ r} and assume there exists

> 0 such that R(z) ≥ 1 + for every z ∈ B. Show using the incomplete version of

Weierstrass M -test that

converges absolutely for every z ∈ B.

Hint: Here n?z = e?z log(n) but this log is the one of real numbers.

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Problem 4. (10 pts) Let F ∶ A × [0,1] where A is open and connected. Suppose that

z → F (z, s) is holomorphic for everywhere on A for every s.

F is continuous in A × [0,1] and

¨

A×[0,1]

F (z, s)dsdz <∞

Define f ∶ A→ C via

f(z) = ? 1

0

F (z, s)ds

1. (8 pts) Show that f is holomorphic on A. Be sure to justify every step.

2. (2 pts) Show that

H(z) = ? 1

0

e?z2t2dt

is entire and compute H ′(z).

Hint for 1: What are the hypothesis for Fubini’s theorem?

For 2: One unjustified step is allowed as long as you clearly identify what step you are

not justifying.

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Problem 5. (10 pts) A set A ? C is said to be symmetric with respect to the x?axis if z ∈ A

if and only if z ∈ A.

Let A be symmetric with respect to the x-axis, denote

A+ = A ∩ {z ∶ I(z) > 0} ,

A = A ∩ {z ∶ I(z) < 0},

I = A ∩ {z ∶ I(z) = 0}.

Observe that A = A+ ∪ I ∪A?.

Suppose f ∶ A+ → C is holomorphic and there exists a continuous function g ∶ A+ ∪ I → C such

that

gA+ = f ,

g(I) R.

Show there exists a function F ∶ A→ C such that F is holomorphic on all A.

Hint: For A? try F (z) = g(z), what is an ecient way to check holomorphicity? Another

exercise on this midterm may be useful.

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2

Problem 6. (10 pts)

1. (1 pt) Let f be holomorphic on a simply connected, open set A, assume f(z) ≠ 0 for all

z. Let be a closed curve homotopic to a point on A, find the value of

?

f ′(z)

f(z) dz.

Let z0 ∈ A with f(z0) ≠ 0, define L ∶ A→ C via

L(z) = log[0,2?)(f(z0)) + ?

f ′(?)

f(?) d?

where log[0,2?) is the principal branch of logarithm and is any curve in A joining z0

and z.

2. (1 pt) Show that L is well-defined, i.e. independent of the choice of curve .

3. (4 pts) Show that L acts like the logarithm of f , i.e. eL(z) = f(z) for all z ∈ A.

4. (1 pt) Explain how this question relates to the existence of logarithms in simply con-

nected regions (Lecture 6).

5. (3 pts) Show there exists a holomorphic n?th root of f , that is, there exists a holomor-

phic function g such that gn = f .

Notation: In this midterm R(z) and I(z) denote the real and imaginary parts of z re-

spectively.

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