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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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.