Let , and let . What is ?

Let denote the punctured unit disk: . Which of the following is false?

	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.

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Consider the set given by . Which of the following is true?

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

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Given two complex numbers and with principle arguments and and magnitudes and , respectively, which of the following is false?

	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 .

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If satisfies , writing , which of the following statements holds?

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

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Let , and let . What is the principle argument (i.e. the argument in ) of ?

Let , and let . What is the principle argument (i.e. in ) of ?

Suppose that with . What is the modulus of ?

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

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

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Let , with , where and are real-valued. Let denote the ball of radius centered at . Which of the following is true?

	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 .

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

	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

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

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

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Let . Let be an accumulation point of , and let be a complex number. Choose the correct definition for the following statement:

	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 , .

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Which of the following statements about the complex exponential function is false?

	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 .

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

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

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Let be given by . Let . Then which of the following is true?

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

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Let be the Moebius transformation such that , , and . What is ?

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

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What is the modulus of the principle value of ? (Give two decimal places of precision.)

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

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

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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?

	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

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Define the following contours: for .

for .

for .

Which of the following is true?

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

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Let be a function which is analytic on a domain , and let be a simple contour in . Which of the following is true?

	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.

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

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

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Which of the following is true?

	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.

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Let be the positively oriented circle of radius centered at in the complex plane. What is the value of ?

Let be the positively oriented circle of radius centered at in the complex plane. What is the value of ?

Compute , where is the line segment from to . (Give precision to the hundredths place.)

Let . Let . Let . Which of the following is false?

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

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Consider the power series . Which of the following is false?

	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 .

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

Suppose a sequence of continuous functions converges uniformly to zero on . True or false: .

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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.

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Consider the function . Denote the punctured unit disc by . Let and . Which of the following statements is false?

	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.

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Consider the function on . Which of the following is true about the singularities of ?

	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 .

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Consider the function . Which of the following statements is true?

	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 .

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

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

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Let and be non-constant holomorphic functions on with . Let be a bounded domain. Which of the following is possible?

	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 .

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Let be a meromorphic function defined on by . Let be parameterized by , . What is the value of ?

Calculate the residue of at .

Calculate the residue of at .

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?

	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.

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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?

	A. .
	B. .
	C. is constant inside .
	D. has no zeros inside or on for .
	E. has a power series representation valid inside .

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

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

Let . Let be parameterized by , . What is the value of ?

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

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

Let be an open set. Let . Which of the following is true?

	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 .

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Let be a simply connected, open set. Let be an entire function, and let on . Which of the following is the strongest statement that can be made about ?

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

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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 .

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Let be a map which is conformal from to , where . Which of the following is a possible choice for ?

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

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Let . If , what is ?

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

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Let . Define a mapping on by . What is ?

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

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Suppose that is a simply connected, open set. Let be a nonconstant function. Which of the following is true?

	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.

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Let , and let . If an analytic bijection does not exist between and , which of the following is not a possible cause?

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

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