Let

, and let

. What is

Answer:

Question 2

Let

denote the punctured unit disk:

. Which of the following is false?

Choose one answer.

	A. 0 is an isolated point of .
	B. 0 is a boundary point of .
	C. 0 is an accumulation point of .
	D. is open.
	E. is connected.

Question 3

Consider the set

given by

. Which of the following is true?

Choose one answer.

	A. is closed.
	B. has no interior points.
	C. has no boundary points.
	D. has no isolated points.
	E. has no accumulation points.

Question 4

Given two complex numbers

and

with principle arguments

and

and magnitudes

and

, respectively, which of the following is false?

Choose one answer.

	A. The magnitude of is .
	B. The principle argument of is .
	C. The magnitude of does not exceed .
	D. If , corresponds to a rotation of by an angle of .
	E. If , then corresponds to a dilation of by .

Question 5

satisfies

, writing

, which of the following statements holds?

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. None of these choices

Question 6

Let

, and let

. What is the principle argument (i.e. the argument in

) of

Answer:

Question 7

Let

, and let

. What is the principle argument (i.e. in

) of

Answer:

Question 8

Suppose that

with

. What is the modulus of

Answer:

Question 9

Which of the following pairs of functions fails to satisfy the Cauchy-Riemann equations?

Choose one answer.

	A. , .
	B. , .
	C. , .
	D. , .
	E. , .

Question 10

Let

, with

, where

and

are real-valued. Let

denote the ball of radius

centered at

. Which of the following is true?

Choose one answer.

	A. In order for to be differentiable on , it is necessary that the Cauchy-Riemann equations hold for the partial derivatives of and on .
	B. In order for to be differentiable at , it is sufficient that the Cauchy-Riemann equations hold for the partial derivatives of and at .
	C. In order for to be differentiable at , it is necessary and sufficient that the Cauchy-Riemann equations hold for the partial derivatives of on for some .
	D. In order for to be differentiable at , it is neither necessary nor sufficient that the Cauchy-Riemann equations hold for the partial derivatives of on for some .
	E. In order for to be differentiable at , it cannot be the case that the Cauchy-Riemann equations hold for the partial derivatives of on for any .

Question 11

Which of the following is true of the complex-valued function

Choose one answer.

	A. is not continuous.
	B. is continuous, but nowhere holomorphic.
	C. is holomorphic at some points in , but not entire.
	D. is an entire function.
	E. None of these choices

Question 12

Let

be the real part of an analytic function. Which of the following is the harmonic conjugate,

, of

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. None of these choices

Question 13

Let

. Let

be an accumulation point of

, and let

be a complex number. Choose the correct definition for the following statement:

Choose one answer.

	A. , there exists such that satisfying , .
	B. , there exists such that satisfying , .
	C. , there exists such that satisfying , .
	D. , there exists such that satisfying , .
	E. , there exists such that satisfying , .

Question 14

Which of the following statements about the complex exponential function

is false?

Choose one answer.

	A. The complex exponential is periodic.
	B. The complex exponential is onto (i.e. surjective).
	C. The complex exponential satisfies .
	D. The complex exponential is holomorphic.
	E. The complex exponential satisfies for all .

Question 15

Let

be the principle value of the logarithm. Which of the following could be the preimage of the set

Choose one answer.

	A. The annulus .
	B. The annulus .
	C. The punctured disc .
	D. The punctured disc .
	E. None of these choices

Question 16

Let

be given by

. Let

. Then which of the following is true?

Choose one answer.

	A. is a circle.
	B. is a line.
	C. is an ellipse.
	D. is a point.
	E. is the punctured unit disc.

Question 17

Let

be the Moebius transformation such that

, and

. What is

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. .

Question 18

What is the modulus of the principle value of

? (Give two decimal places of precision.)

Answer:

Question 19

True or false: A well-defined principle branch of the complex logarithm is an entire function.

Answer:

True False

Question 20

True or false: There exists a Moebius transformation which has derivative equal to zero at the origin.

Answer:

True False

Question 21

Suppose

is a function on an open set

. Let

be a piecewise smooth, simple curve in

Let

be a smooth parametrization of

. Which of the following is true?

Choose one answer.

	A. If is continuous on and is closed, then for all on the interior of .
	B. If is continuous and is closed, then .
	C. If is holomorphic and is a primitive for , then .
	D. If is holomorphic, then its derivatives are bounded on .
	E. None of these choices

Question 22

Define the following contours:

for

.

Which of the following is true?

Choose one answer.

	A. is simple.
	B. is closed.
	C. is homotopic to in .
	D. is homotopic to in .
	E. The winding number of around is zero.

Question 23

Let

be a function which is analytic on a domain

, and let

be a simple contour in

. Which of the following is true?

Choose one answer.

	A. In order to have , it is necessary that is closed.
	B. In order to have , it is sufficient that is closed.
	C. In order to have , it is necessary and sufficient that is closed.
	D. In order to have , it is neither necessary nor sufficient that is closed.
	E. In order to have , it cannot be the case that is closed.

Question 24

Which of the following smooth curves in the complex plane is longest?

Choose one answer.

	A. , .
	B. , .
	C. , .
	D. , .
	E. , .

Question 25

Which of the following is true?

Choose one answer.

	A. Analytic functions which are bounded on a subset of their domain are constant on that subset.
	B. is a bounded, entire function.
	C. If is a well-defined branch of the logarithm and is a simple closed contour in , then .
	D. If is a polynomial of degree on , then if is a bounded, entire function, we know with certainty.
	E. If a function satisfies for all simple closed contours in , then is bounded.

Question 26

Let

be the positively oriented circle of radius

centered at

in the complex plane. What is the value of

Answer:

Question 27

Let

be the positively oriented circle of radius

centered at

in the complex plane. What is the value of

Answer:

Question 28

Compute

, where

is the line segment from

. (Give precision to the hundredths place.)

Answer:

Question 29

Let

. Let

. Which of the following is false?

Choose one answer.

	A. converges to on .
	B. converges uniformly to on .
	C. converges to 1.
	D. is constant.
	E. converges absolutely on .

Question 30

Consider the power series

. Which of the following is false?

Choose one answer.

	A. The radius of convergence is at least 1 if is bounded.
	B. The radius of convergence is at most 1 if does not converge to zero.
	C. If is strictly increasing, then the series does not converge.
	D. If there exists with for all and for some , has radius of convergence , then has radius of convergence of at least .
	E. If exists, then is the radius of convergence of .

Question 31

What is the radius of convergence for the following power series in

Answer:

Question 32

Suppose a sequence of continuous functions

converges uniformly to zero on

. True or false:

Answer:

True False

Question 33

Suppose a function

has a Laurent series of the form

. Match the type of function with its Laurent series.

for every .
A nonzero, finite number of the coefficients with are nonzero.
An infinite number of the coefficients with are nonzero.

Question 34

Consider the function

. Denote the punctured unit disc by

. Let

and

. Which of the following statements is false?

Choose one answer.

	A. Given any point in , there is a point s.t. with .
	B. is not bounded on .
	C. is analytic on .
	D. is an entire function.
	E. has no poles.

Question 35

Consider the function

. Which of the following is true about the singularities of

Choose one answer.

	A. has an essential singularity at .
	B. has a removable singularity at .
	C. has an essential singularity at .
	D. has a pole at .
	E. has poles at and .

Question 36

Consider the function

. Which of the following statements is true?

Choose one answer.

	A. has a removable singularity at .
	B. has a removable singularity at .
	C. has an essential singularity at .
	D. has a pole at .
	E. has a pole at .

Question 37

The proof that holomorphic functions on open, connected sets have power series representations relies on which of the following results?

Choose one answer.

	A. Morera's theorem
	B. Cauchy's integral formula
	C. Poisson's integral representation
	D. Liouville's theorem
	E. Riemann mapping theorem

Question 38

Let

and

be non-constant holomorphic functions on

with

. Let

be a bounded domain. Which of the following is possible?

Choose one answer.

	A. attains a maximum on the interior of G.
	B. for all , where is sequence in converging to a point in .
	C. and have the same power series representations on .
	D. is a bounded function.
	E. There exists a contour in such that .

Question 39

Let

be a meromorphic function defined on

. Let

be parameterized by

. What is the value of

Answer:

Question 40

Calculate the residue of

Answer:

Question 41

Calculate the residue of

Answer:

Question 42

Let

be a function on an open set

with singularities at

. Let

be a closed curve in

. Which of the conditions below is not necessary for the following formula to hold?

Choose one answer.

	A. is holomorphic on .
	B. is simply connected.
	C. is positively oriented and simple.
	D. The singularities lie inside .
	E. All of these choices are necessary.

Question 43

Suppose

is a function that is holomorphic inside and on a simple, closed curve

, and that for all

inside and on

. Which of the following is false?

Choose one answer.

	A. .
	B. .
	C. is constant inside .
	D. has no zeros inside or on for .
	E. $\frac{1}{f(z)$ has a power series representation valid inside .

Question 44

Let

. Let

be the positively oriented circle of radius 2, centered at the origin. What is the value of

Answer:

Question 45

Let

be the positively oriented circle of radius 2, centered at the origin. What is the value of

Answer:

Question 46

Let

. Let

be parameterized by

. What is the value of

Answer:

Question 47

Compute

. (Hint: use residues.) Your answer should be correct to the hundredths place.

Answer:

Question 48

Compute

. (Hint: use a half-annulus contour.) Your answer should be correct to the hundredths place.

Answer:

Question 49

Let

be an open set. Let

. Which of the following is true?

Choose one answer.

	A. In order for to be harmonic, it is necessary that there exist a holomorphic function on with .
	B. In order for to be harmonic, it is sufficient that there exist a holomorphic function on with .
	C. In order for to be harmonic, it is necessary and sufficient that there exist a holomorphic function on with .
	D. In order for to be harmonic, it is neither necessary nor sufficient that there exist a holomorphic function on with .
	E. In order for to be harmonic, there cannot exist a holomorphic function on with .

Question 50

Let

be a simply connected, open set. Let

be an entire function, and let

. Which of the following is the strongest statement that can be made about

Choose one answer.

	A. is constant on .
	B. on .
	C. is constant on .
	D. has no zeros in .
	E. has infinitely many zeros in .

Question 51

True or false: Consider a function

, where

and

are real-valued. Let

be the annulus

, and let

be the disc

. Suppose that

achieves its maximum in

on the interior of

. Then

cannot be holomorphic in

Answer:

True False

Question 52

Let

be a map which is conformal from

, where

. Which of the following is a possible choice for

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. , where is the principal branch of the logarithm.

Question 53

Let

. If

, what is

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. .

Question 54

Let

. Define a mapping on

. What is

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. .

Question 55

Suppose that

is a simply connected, open set. Let

be a nonconstant function. Which of the following is true?

Choose one answer.

	A. In order for to be open for any open subset , it is necessary that be analytic.
	B. In order for to be open for any open subset , it is sufficient that be analytic.
	C. In order for to be open for any open subset , it is necessary and sufficient that be analytic.
	D. In order for to be open for any open subset , it is neither necessary nor sufficient that be analytic.
	E. In order for to be open for any open subset , cannot be analytic.

Question 56

Let

, and let

. If an analytic bijection does not exist between

and

, which of the following is not a possible cause?

Choose one answer.

	A. is not open.
	B. is not simply connected.
	C. .
	D. contains the origin.
	E. Both A and B