Qualifying Exam–Complex Analysis (Math 530)–August 2019

Notations: D := z ∈ C : |z| < 1 and D := z ∈ C : |z| ≤ 1.

1.(a) Suppose that u(x, y) is a harmonic function on a domain D and that v(x, y) is itsharmonic conjugate to prove that u(x, y)v(x, y) is harmonic on D.

(b) Give an example to two harmonic functions u(x, y) and v(x, y) defined on a domain Dsuch that u(x, y)v(x, y) is not harmonic on D.

2.(a) Determine the radius of convergence for∞∑n=1

n2zn.

(b) What function does the power series from Part (a) represent.

(c) Use your solution to Part (b) to explain why the radius of convergence that you foundin Part (a) is correct.

3. Compute

∫ ∞−∞

x2

(x2 + 9)2dx by using residus. Please give complete proofs of any esti-

mates that you use.

4. Suppose that f : D→ D is holomorphic and that there are two distinct points z1, z2 ∈ Dwith

f(z1) = z1 = f(z2).

Prove that |f ′(z1)| < 1.

5. Prove that for any ε > 0 the function f(z) = 1z+i

+ sin(z) has infinitely many zeros inthe strip z ∈ C : −ε < Im(z) < ε.

6. Let f : U → C be a non-constant holomorphic function defined on an open set Ucontaining the closed unit disc D. Prove that if |f(z)| = 1 whenever |z| = 1, then f(U)contains the open unit disc D.

Hint: first show that 0 ∈ f(U). Then use suitably chosen Mobius transformationsm : D→ D to prove the full statement.

1

Qualifying Exam–Complex Analysis (Math 530)–January 2019

1. Let D be a domain. A function f : D → C is antiholomorphic if for every z ∈ D the limit

limh→0

f(z + h)− f(z)

h

exists. Write f(z) = f(x+ iy) = u(x, y) + iv(x, y) with x, y ∈ R and derive an analog ofthe Cauchy Riemann Equations that an antiholomorphic function must satisfy.

2. Compute

∫ ∞0

dx

x3 + 1.

Hint: integrating 1/(z3 + 1) around the boundary of the circular sectorz = reiθ : 0 ≤ θ ≤ 2π/3, 0 ≤ r ≤ ρ and let ρ→∞.

3. (a) Determine the Laurent Series expansion for f(z) = 1(z−1)(z−2i) that converges in the

annulus where 1 < |z| < 2.

(b) Determine the first four terms of the Laurent Series Expansion of g(z) = 1ez−1 in the

annulus 0 < |z| < 2π.

4. Let Ω := z ∈ C : |z| > 4.

(a) Show that there is an analytic function defined on Ω whose derivative is

f(z) =z

(z − 1)(z − 2)(z − 3).

(b) Can there be an analytic function on Ω whose derivative is

g(z) =z2

(z − 1)(z − 2)(z − 3)?

5. Determine all entire functions f : C→ C that satisfy |f(z)| ≤ 1 +√|z| for every z ∈ C.

6. State the Argument Principle and use it to prove the Open Mapping Theorem: Let f bea non-constant analytic function on some domain Ω. Then for any open set U ⊂ Ω wehave that f(U) is open in C.Hint: apply the argument principle to the function f(z)− w.

7. Let p(z) be a complex polynomial of degree equal to 2019 and let

Ω = z ∈ C : |p(z)| < 1.What is the minimum number of connected components of Ω? What is the maximumnumber of connected components of Ω? Give examples of polynomials realizing each.

Hint: Prove that each connected component of Ω contains a zero of p.

1

Qualifying Exam–Complex Analysis (Math 530)–August 2018

Use a separate piece of paper for each problem.

Notations: D := z ∈ C : |z| < 1 and T := z ∈ C : |z| = 1.

1. Let p(z) = z7 + z3 + 116.

(a) Show that each of the zeros of p is simple (multiplicity one).

(b) Determine the number of zeros p has in the closed disc |z| ≤ 12.

2. Use contour integration to compute

∫ ∞−∞

cosx

x2 − 2x+ 10dx.

3. Suppose that f : D → D is a holomorphic function with f(12

)= 1

2and f ′

(12

)= i.

Determine f(−1

2

).

4. Prove that an entire function f : C → C whose values avoid the positive real axis[0,∞) must be a constant function.

(Even if you know it, you may not use the Little Picard Theorem on this problem.)

5. Let U := D \ 0. Suppose f : U → C is a holomorphic function such that for everyz ∈ U we have Re(f) > 0. Prove that 0 is a removable singularity for f .

(Even if you know it, you may not use the Big Picard Theorem on this problem.)

6. It is desired to approximate the function 1/z by a polynomial p(z) on the unit circle T.Prove that that for any p(z) there is a point z0 ∈ T such that |p(z0)− 1/z0| ≥ 1.

In other words, you can’t approximate 1/z very well by any polynomial!

Hint: integrate 1/z − p(z) around the unit circle and explain why your result provesthe claim.

1

Qualifying Exam – Complex Analysis (Math 53000) – January 2018

Notation: D = z ∈ C : |z| < 1, H = z ∈ C : Im(z) > 0.

1. Find a one-to-one analytic map from

z : |z − 1| <√2

∩

z : |z + 1| <√2

onto D.

2. How many zeros does the polynomial z2018 + iz100 + 3 have in the firstquadrant?

3. Let F be the family of all analytic functions f : H → H such thatf(i) = 2018 + 100i. Find sup|f ′(i)| : f ∈ F.

4. Prove or provide a counterexample: There is no entire function f suchthat

f

(

1

n

)

=n

n+ 1

for n = 1, 2, 3, . . ..

5. Let f : z : 0 < |z| < 4 → C be an analytic function. Assume that if|z| = 1 then |f(z)| ≤ 1 and if |z| = 3 then |f(z)| ≤ 9. Prove that if |z| = 2then |f(z)| ≤ 4.

6. Let f be a linear fractional function. Assume that there exists z suchthat f(f(z)) = z, but f(z) 6= z. Prove that f f is the identity.

Qualifying Exam – Complex Analysis (Math 53000) – August 2017

Notation: D = z ∈ C : |z| < 1, H = z ∈ C : Im(z) > 0.

1. Find a one-to-one analytic map from H \ iy ∈ C : y ∈ R, 0 < y ≤ πonto H.

2. How many zeros does the polynomial 2z4 − 6z3 + 8z2 − 6z + 9 have inthe annulus z ∈ C : 1 < |z − 1| < 2?

3. Let f : D → H be an analytic map such that f(0) = i. Determine allpossible values of f(1/2). Try to present your answer in geometrical terms.

4. Prove or provide a counterexample: There is no entire function f suchthat f(R) ⊂ R and f(i) = f(−i) = 2017i.

5. Evaluate∫

∞

0

cosx

x4 + 16dx.

6. Let h : C → R be a continuous function, such that eh is harmonic. Provethat h is a constant function.

Complex Analysis Qualifying ExamIUPUI Winter 2016(exam by R. Perez)

1. (10 points) Prove that the function f(z) = |z2| has a complex derivative atz = 0, and at no other point.

2. (10 points) Compute the improper integral below, justifying your methodology:∫ ∞−∞

x2 dx

x4 + 5x2 + 4.

3. (10 points) Show that if |b| < 1, then the polynomial z3 + 3z2 + bz + b2 hasexactly two roots (counted with multiplicity) in the unit disk.

4. (10 points) Find a conformal map that takes the region|z| < 1 , |z−1/2| > 1/2

in a one-to-one fashion onto the unit disk.

5. (10 points) Prove that if f is entire and satisfies |f(z2)| ≤ |f(z)| for all z ∈ C,then f is constant.

6. (10 points) Show that the function 12

(z + 1/z

)maps the half-disk

|z| <

1 , Im z < 0

in a one-to-one fashion onto the upper half-plane

Im z > 0

.

7. (10 points) Let D denote the unit disk, and suppose the holomorphic functiong : D −→ D has two distinct fixed points. Prove that g(z) ≡ z.

1

QUALIFYING EXAMCOMPLEX ANALYSIS (MATH 53000)

JANUARY 2016

In what follows, D :=z : |z| < 1

and T :=

z : |z| = 1

.

1. Given a circle C = z : |z−a| = R, find two points symmetric with respectto both C and the imaginary axis.

2. Let f be a holomorphic function in D which is continuous in D. Assumingthat |f(z)| > |f(0)| for all z ∈ T, show that f has at least one zero in D.

3. If P is a polynomial of degree n and |P(z)| 6 M for z ∈ D, then |P(z)| 6

M|z|n for z ∈ C \ D.

4. Let f1(z) and f2(z) be functions holomorphic in a domain D. Assume thatboth functions satisfy differential equation f(m) = P(z, f, f(1), . . . , f(m−1)),where P is a polynomial. Show that if f(k)1 (z0) = f

(k)2 (z0) for some z0 ∈ D

and all k ∈ 0, 1 . . . ,m− 1, then f1 ≡ f2 in D.

5. Find∫|z|=2

zndz√1 + z2

.

6. Let f(z) be a univalent function in D such that f(0) = 0 and D := f(D)is symmetric with respect to a line L passing through the origin. Show thatf−1(L ∩D) is a diameter of D.

QUALIFYING EXAMAUGUST 2016

MATH 530 - Prof. Perez

(1) Let a > 0. Find the shortest and longest distances from the origin to the curve∣∣z + 1z

∣∣ = a.

(2) Let C be the (positively oriented) boundary of the half-ring pictured. Find theintegral ∫

C

z

zdz.

210−1−2

(3) Show that for any radius r > 0 there exists N ≥ 1 such that for all n ≥ N thepolynomial

1 + z + z2

2!+ . . .+ zn

n!

has no roots within the disk of radius r centered at the origin.

(4) Find a one-to-one conformal mapping of the upper half plane without the verticalray [i, i∞), onto the unit disc. (The answer can be given as a sequence of maps)

(5) Let f be a holomorphic function defined in a neighborhood of 0, and such thatf(0) = 0. We say that 0 is a fixed point of f . In complex dynamics, two importantinvariants of such a fixed point are its multiplier λ := f ′(0) and its fixed point index

ι := 12πi

∫C

dz

z − f(z),

where C is a small, positively oriented curve around 0.

Prove that whenever λ 6= 1, these invariants are related by the formula

ι =1

1− λ.

(6) Evaluate the real integral ∫ ∞

0

2x2 − 1

x4 + 5x2 + 4dx.

QUALIFYING EXAMCOMPLEX ANALYSIS (MATH 53000)

AUGUST 2015

In what follows, D :=z : |z| < 1

.

1. Find the reflection of the circle|z− a| = r

across the circle

|z| = R

.

2. Let f be a holomorphic function in D which is continuous in D. Assumingthat |f(z)| < 1 for all z ∈ D, show that there exists a unique z0 ∈ D such thatf(z0) = z0.

3. Let D be a simply connected domain andfn

a locally uniformly boundedsequence of univalent functions in D. Assuming that f

(k)n (z0) → wk as

n → ∞ for some z0 ∈ D and all k > 0 and that wk are not all zero fork > 1, show that

fn

converges locally uniformly to a univalent functionin D.

4. Let f be a holomorphic function in D except for a simple pole at z0 6= 0.Show that an/an+1 → z0 as n →∞, where f(z) =

∑∞n=0 anz

n.

5. Compute∫π

0

(sin(nx)sin(x)

)dx.

6. Show that1

1 − z=

∞∏n=0

(1 + z2n) , z ∈ D.

Qualifying Exam — Complex Analysis (Math 53000)

January 2014

V.Tarasov

1. Describe and sketch the set Re(tan z) > 0 .

2. The region D is the intersection of the disks |z − 1| <√

2 and |z + 1| <√

2 .Construct a conformal mapping (one-to-one) from D to the unit disk |z | < 1 .First represent the mapping as a composition of elementary mappings, and thensimplify the final formula as much as possible.

3. Let f(z) be an entire function that does not take negative real values.Show that f(z) is a constant function.

4. The function f(z) is analytic in the unit disk |z | < 1 and maps it surjectively to the

upper half plane Re z > 0 . Find the radius of convergence of the Taylor series∞∑

k=0

akzk

of the function f(z) .

5. a) How many solutions does the equation 2z2 = cos z have in the unit disk |z | < 1 ?b) Show that all solutions of the equation 2z2 = cos z that lie in the unit disk |z | < 1

are real.

6. Evaluate

∞∫

0

sin(πx)

x − x3dx . Explain each step.

Math 53000 Qualifying Exam August 2014

Notation: unit disk: D = z : |z| < 1 and right half plane: H+ = z : Re(z) > 0.

1. (a) Give a careful statement of the Schwarz Lemma, including information about the derivative andequality in the inequalities, which begins: If ϕ : D 7→ D is an analytic function mapping the unitdisk into itself such that ϕ(0) = 0, then · · ·

(b) Let f be an analytic function mapping the right half plane, H+, into itself such that f(1) = 1.Prove that for w in H+,∣∣∣∣f(w)− 1

f(w) + 1

∣∣∣∣ ≤ ∣∣∣∣w − 1

w + 1

∣∣∣∣(c) What can you say about f if there is w0 6= 1 for which equality holds in this inequality?

2. Find the number of roots of the polynomial p(z) = z7 − 8z2 + 2 in the annulus z : 1 < |z| < 2.Prove that your answer is correct, citing appropriate theorems and justifying their hypotheses.

3. Let S = z : Re(z) > 0 and |Im(z)| < 3π

2 and let g(z) = ez for z in S.

(a) Describe the set g(S) = w ∈ C : w = g(z) for some z in S.(b) For any set F , let #(F ) denote the number of elements of F , so #(∅) = 0, #(C) = ∞, and

#(n : n is an integer and |n| ≤ 10) = 21.

For each point w in g(S), find #(z ∈ S : g(z) = w), that is, for each w in g(S), find thenumber of points of S that map to w.

4. Let Ω1 = z = x+ iy : −3 < x < 2 and y = 0 ∪ z = x+ iy : x < 2 and y 6= 0and Ω2 = z = x+ iy : −3 < x < 2 and y = 0 ∪ z = x+ iy : x > −3 and y 6= 0.

The function ζ is analytic on Ω1 and has a non-removable singularity at −3. Similarly, the function ηis analytic on Ω2 and has a non-removable singularity at 2.

Finally, these functions satisfy ζ(−1) = η(−1) and the derivatives ζ(k)(−1) = η(k)(−1) for allpositive integers k, where h(1)(z) = h′(z), h(2)(z) = h′′(z), etc.

(a) Find the coefficients ak (in terms of information about ζ) so that the power series for ζ centeredat c = −1 is

∑∞k=0 ak(z + 1)k.

(b) What is the radius of convergence of the power series for ζ centered at a = −1? Explain why!

(c) Prove that ζ(z) = η(z) for all z in Ω1 ∩ Ω2 = z = x+ iy : −3 < x < 2.

5. Find the Laurent series for h(z) =z

z2 + 2z − 3that converges on the open annulus z : 1 < |z| < 3.

6. Use contour integration to compute

∫ ∞

0

x1/3

1 + x2dx.

Justify your calculation by explicitly describing the contour(s), the substitutions used in each pieceof the contour(s), and the estimates necessary for drawing your conclusion. (You should express youranswer in a form that is obviously a real number!)

Math 53000 – Qualifying Examination – January 2013

Notation: 𝔻 = 𝑧 ∈ ℂ : ∣𝑧∣ < 1, ℍ = 𝑧 ∈ ℂ : Im(𝑧) > 0.

1. Find the number of zeros of the function

𝑓(𝑧) = 2013𝑧8 + 1000 + 𝑒(3+4𝑖)𝑧

in 𝔻.

2. Evaluate

∫ ∞

0

cos𝑥

𝑥2 + 4𝑑𝑥 .

3. Find a one-to-one analytic map from

𝑧 ∈ ℂ : ∣𝑧∣ < 1, Re(𝑧) > 0, Im(𝑧) > 0 ∖ 𝑎(1 + 𝑖) : 𝑎 ∈ (0, 1/2]onto ℍ. You may represent your map as a finite composition of maps.

4. Find all analytic maps 𝑓 : ℍ → ℍ for which 𝑓(𝑖) = 𝑖 and 𝑓 ′(𝑖) = −1.

5. Consider the following statements:

(a) If 𝑓 : ℂ → ℂ is an analytic function then the function 𝑔 : ℂ → ℂ, givenby 𝑔(𝑧) = 𝑓(𝑧), is also analytic.

(b) If 𝑢 : ℂ → ℝ is a harmonic function then the function 𝑣 : ℂ → ℝ, givenby 𝑣(𝑧) = 𝑢(𝑧), is also harmonic.

For each statement, decide whether it is true or false. If it is true, prove it;if it is false, give a counterexample.

6. Let 𝑓 : 𝔻 → ℂ be an analytic function such that if ∣𝑧∣ = 1/2 then𝑓(𝑧) ∈ ℝ. Prove that 𝑓 is constant.

Qualifying Exam — Complex Analysis (Math 53000)

August 2013

V.Tarasov

1. Describe and sketch the set Im(z3 + z−3) < 0 .

2. The region D = z ∈ C : |z − 1| > 1 , |z + 1| > 1 is the complement of the union oftwo disks. Construct a conformal mapping (one-to-one) from D to the unit disk |z | < 1 .First represent the mapping as a composition of elementary mappings, and then simplifythe final formula as much as possible.

3. Find all entire functions f(z) such that |f(z)| = 1 for |z | = 1 .Hint : Show first that f(z) is a polynomial.

4. The series∞∑

k=0

akzk converges in the unit disk |z | < 1 and defines a function mapping

the unit disk into itself. Show that |a0 |2 + |a1| 6 1 .

5. Show that the equation z = tan(πz) has exactly three solutions in the strip |Re z | < 1 .Hint : Keep in mind the proof of Rouche’s theorem.

6. Evaluate

∞∫

0

cos(πx/2)

1 − x4dx . Explain each step.

Math 53000 – Qualifying Examination – January 2012

Notation: 𝔻 = 𝑧 ∈ ℂ : ∣𝑧∣ < 1.

1. Find the number of zeros of the polynomial

𝑃 (𝑧) = 𝑧2012 + 𝑧20 + 𝑧12 + 𝑖

in the second quadrant.

2. Evaluate the integral ∫ ∞

0

sin 𝑥 𝑑𝑥

𝑥3 + 𝑥.

3. Find all entire functions 𝑓 such that

∣𝑓(𝑧)∣ < ∣𝑧∣+ 2012

for all 𝑧 ∈ ℂ.

4. Find a conformal map from 𝔻 onto 𝔻∖ [−12 , 1

). You may represent your

map as a finite composition of maps.

5. Let 𝑃 be a polynomial with real coefficients. Prove that there is aharmonic function ℎ : ℂ → ℝ such that ℎ(𝑖𝑦) = 𝑃 (𝑦) for all 𝑦 ∈ ℝ.

6. Let 𝑓 : 𝔻 → ℂ be an analytic function such that 𝑓(0) = 0 and

sup𝑧∈𝔻

∣𝑓(𝑧)∣ < 1.

Define by induction: 𝑧1 = 12 , 𝑧𝑛+1 = 𝑓(𝑧𝑛) for 𝑛 = 1, 2, . . .. Prove that

lim𝑛→∞ 𝑧𝑛 = 0.

Qualifying Exam — Complex Analysis (Math 53000)

January 2011

V.Tarasov

1. Describe and sketch the set Re(tan2z) > 0 .

2. Find all analytic functions f(z) defined in the upper half plane z ∈ C : Im z > 0such that |f(z)| < 1 for all z , f(i) = 0 and f ′(i) = 1/2 .

3. Let a polynomial P (z) = 1 + a1z + . . . + anzn , n ≥ 1 , be such that |P (z)| ≥ 1for all z on the circle |z | = 1 . Show that P (z) has a root in the disk |z | < 1 .

4. Find all zeros and singularities of the functione1/z sin z

1 − cos zincluding infinity. Classify the

singularities. For each zero and pole, find the order and compute the residue at the pole.

5. Evaluate

∫

|z|=1

dz

sin(4/z)for the positive sense of the circle.

6. Let f(z) be the branch of log z such that 0 6 Im f(z) < 2π . Write the Taylor seriesfor f(z) at z = 1−i . Determine the radius of convergence of the Taylor series. Expressthe sum of the Taylor series series in terms of f(z) .

7. Evaluate

∞∫

0

1 − cos x

x2dx .

Math 53000 – Qualifying Examination – August 2011

1. Find the number of zeros of the polynomial

𝑃 (𝑧) = 𝑧2011 − 1000𝑧1000 − 2011in the annulus 𝑧 ∈ ℂ : 1 < ∣𝑧∣ < 2.

2. Evaluate the integral ∫ ∞

−∞

𝑑𝑥

𝑥2 + 14𝑥+ 113.

Explain all steps.

3. Let 𝐷 = 𝑧 : ∣𝑧∣ < 1 be the unit disk. Does there exist an analyticfunction 𝑓 : 𝐷 → ℂ such that

𝑓

(1

𝑛

)= 𝑓

(− 1𝑛

)=1

𝑛3

for 𝑛 = 2, 3, 4, . . .? Give an example of such a function or prove that it doesnot exist.

4. Find a conformal map from the set

𝑧 ∈ ℂ : ∣𝑧∣ > 1 ∖ (−∞,−1)onto the set

ℂ ∖ (−∞, 0].

You may represent your map as a finite composition of maps.

5. Find all singular points of the function

𝑓(𝑧) =(sin(𝑧3))7

𝑧21

and classify them.

6. Find all harmonic functions 𝑓 : ℂ ∖ 0 → ℝ that are constant on everycircle centered at 0.

Qualifying Examination – Math 53000 – 1/4/2010

Notation:D = z ∈ C : |z| < 1, D = z ∈ C : |z| ≤ 1, H = z ∈ C : Im z ≥ 0 .

1. Evaluate∫ ∞

0

cos x

(x2 − 1)(x2 − 9)dx.

2. Does there exist an analytic function f , defined in a neighborhood of D,such that f(D) = H? Give an example or prove that it does not exist.

3. Prove that there is no entire function f such that |f(z)| > |z| for allz ∈ C.

4. Find all singular points in C and residues at those points of the function

f(z) =z4

(z2 − 4)3.

5. Write the Laurent series of the function

f(z) =z2010

(z − 1)(z + 2)

in the annulus 1 < |z| < 2.

6. Find all analytic functions f : D → D such that f(0) = 1/3 andf(1/3) = 0.

Qualifying Exam — Complex Analysis (Math 53000)

August 2010

1. Describe and sketch the set Im(z − 1)3

(z + 1)3> 0 .

2. Find all analytic functions f(z) defined in the unit disk z ∈ C : |z | < 1 such thatIm f(z) > 0 for all z , f(0) = i and f(1/3) = 2i .

3. Let f(z) be an analytic function in the disk z ∈ C : |z | < R , where R > 1 .

Prove that

∫

|z|=1

f(z) dz = 2πi f ′(0) . The integral is for the positive sense of the circle.

4. Let f(z) =log(z2 + 1)

(z + 2)2, where −π 6 Im log(z2 + 1) < π . Write the Laurent series

for f(z) around z = −2 . Determine if the obtained series is convergent at z = 1 , z = 0 ,

z = −1 , and find the sum of the series at the points where the series is convergent.

5. Define by induction: a0 = 1/2 , a1 = 1/3 , and an = an−1 an−2 for n > 2 . Prove that

the series f(z) =∞∑

n=0

anzn is convergent for all z . Show that there exists a sequence

z1, z2, . . . such that limn→∞

zn = ∞ and limn→∞

f(zn) = 0 .

6. Evaluate

∞∫

0

x√

x

x3 + 1dx using residues. Simplify the answer as much as possible.

Qualifying Examination – Math 53000 – 8/17/2009

Notation: D = z ∈ ℂ : ∣z∣ < 1, ℍ = z ∈ ℂ : Im z > 0.

1. Evaluate

∫ ∞

0

x2 + 1

x4 + 1dx.

2. How many zeros does the polynomial z9 + z4 + 4 have in the firstquadrant? Explain.

3. Let ℱ be the set of all analytic functions f : ℍ → ℍ such thatf(i) = 2009 i. Find sup

f∈ℱ∣f ′(i)∣.

4. What are the possible types of singularity of the product fg, iff, g : D ∖ 0 → ℂ are analytic functions, f has an essential singularityat 0, and:a. g has an essential singularity at 0?b. g has a pole at 0?

Explain.

5. Define by induction: a0 = 2i, an+1 = a2n. Find the radius of convergence

of the series∞∑

n=0

anzn!.

6. Find the image of D under the map f(z) = z2 − 1

z2.

Qualifying Examination (Complex Analysis)IUPUI, August 2008 – Rodrigo A. Perez

1. Find a on-to-one conformal mapping from the upper half-plane without the segment(0, i] onto the unit disk (the answer can be given as a series of maps).

2. (a) Determine the number of points z in the unit disk such that ez = 3z4.

(b) Determine the number of points z in the complex plane such that ez = 3z4.

3. Use contour integration to compute the integral∫∞

−∞

dx

x2 + 2x + 2.

4. Suppose f(z) is an entire fuction such that |f(z)| ≤ A|z| for all z, where A is somefixed positive number. Show that f(z) = az, where a is a complex constant.

5. Compute the Laurent series representation ofez

(z + 1)2around z = −1 and show that

it is valid for 0 < |z + 1| < ∞.

6. Let γ be the circle |z| = 2 described counterclockwise. Evaluate the integral∫γ

5z − 2

z(z − 1)dz

(a) by finding the residues at the finite poles,

(b) by finding the residue at ∞.

7. Compute1

2πi

∫γ

z3 + z

z − z0

dz,

where z0 stands for each of −1, 0, 1, i and γ is the curve below

i

−1 0 1

8. Let P (z) =∑n

k=0akz

k be a polynomial of degree n. Prove that if |a0| < |an|, then P

has at least one zero inside the unit disk.

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