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Complex Analysis

This quiz covers various concepts and techniques in Complex Analysis.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is the imaginary unit?

  1. $i$
  2. $\pi$
  3. $\sqrt{-1}$
  4. $\infty$
Question 2 Multiple Choice (Single Answer)

What is the Cauchy-Riemann equation in complex analysis?

  1. $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
  2. $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
  3. $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
  4. $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
Question 3 Multiple Choice (Single Answer)

What is the residue of a function at a pole?

  1. The value of the function at the pole
  2. The coefficient of the highest power of $z - a$ in the Laurent expansion of the function around the pole $a$
  3. The integral of the function around a small circle centered at the pole
  4. The derivative of the function at the pole
Question 4 Multiple Choice (Single Answer)

Which of the following is a consequence of Cauchy's Integral Theorem?

  1. The fundamental theorem of calculus
  2. The residue theorem
  3. The maximum modulus principle
  4. Liouville's theorem
Question 5 Multiple Choice (Single Answer)

What is the maximum modulus principle?

  1. A theorem that states that the maximum value of a continuous function on a closed and bounded set is attained at a boundary point.
  2. A theorem that states that the maximum value of an analytic function on a closed and bounded set is attained at a boundary point.
  3. A theorem that states that the maximum value of a harmonic function on a closed and bounded set is attained at a boundary point.
  4. A theorem that states that the maximum value of a subharmonic function on a closed and bounded set is attained at a boundary point.
Question 6 Multiple Choice (Single Answer)

What is the argument principle?

  1. A theorem that relates the number of zeros of an analytic function inside a closed contour to the change in the argument of the function around the contour.
  2. A theorem that relates the number of poles of an analytic function inside a closed contour to the change in the argument of the function around the contour.
  3. A theorem that relates the number of zeros and poles of an analytic function inside a closed contour to the change in the argument of the function around the contour.
  4. A theorem that relates the number of zeros of an analytic function outside a closed contour to the change in the argument of the function around the contour.
Question 7 Multiple Choice (Single Answer)

What is the residue theorem?

  1. A theorem that relates the value of a complex function at a point to the integral of the function around a closed contour.
  2. A theorem that relates the value of a complex function at a pole to the integral of the function around a small circle centered at the pole.
  3. A theorem that relates the value of a complex function at a zero to the integral of the function around a small circle centered at the zero.
  4. A theorem that relates the value of a complex function at a singularity to the integral of the function around a small circle centered at the singularity.
Question 8 Multiple Choice (Single Answer)

What is the order of a pole of a complex function?

  1. The degree of the denominator of the function at the pole
  2. The degree of the numerator of the function at the pole
  3. The degree of the function at the pole
  4. The degree of the derivative of the function at the pole
Question 9 Multiple Choice (Single Answer)

What is the Laurent expansion of a complex function around a point?

  1. An infinite series representation of the function in terms of powers of $z - a$
  2. An infinite series representation of the function in terms of powers of $z$
  3. An infinite series representation of the function in terms of powers of $1/z$
  4. An infinite series representation of the function in terms of powers of $1/(z - a)$
Question 10 Multiple Choice (Single Answer)

What is the principle of analytic continuation?

  1. A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still analytic.
  2. A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still continuous.
  3. A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still differentiable.
  4. A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still integrable.
Question 11 Multiple Choice (Single Answer)

What is the Riemann mapping theorem?

  1. A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the unit disk.
  2. A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the upper half-plane.
  3. A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the lower half-plane.
  4. A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the right half-plane.
Question 12 Multiple Choice (Single Answer)

What is the Schwarz-Christoffel formula?

  1. A formula that gives the conformal mapping of a polygon onto the upper half-plane.
  2. A formula that gives the conformal mapping of a polygon onto the unit disk.
  3. A formula that gives the conformal mapping of a polygon onto the lower half-plane.
  4. A formula that gives the conformal mapping of a polygon onto the right half-plane.
Question 13 Multiple Choice (Single Answer)

What is the Weierstrass factorization theorem?

  1. A theorem that states that every entire function can be written as a product of a sequence of elementary functions.
  2. A theorem that states that every meromorphic function can be written as a product of a sequence of elementary functions.
  3. A theorem that states that every analytic function can be written as a product of a sequence of elementary functions.
  4. A theorem that states that every harmonic function can be written as a product of a sequence of elementary functions.
Question 14 Multiple Choice (Single Answer)

What is the Mittag-Leffler expansion theorem?

  1. A theorem that states that every meromorphic function can be written as a sum of a series of partial fractions.
  2. A theorem that states that every analytic function can be written as a sum of a series of partial fractions.
  3. A theorem that states that every harmonic function can be written as a sum of a series of partial fractions.
  4. A theorem that states that every entire function can be written as a sum of a series of partial fractions.
Question 15 Multiple Choice (Single Answer)

What is the Runge approximation theorem?

  1. A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of rational functions.
  2. A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of polynomials.
  3. A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of trigonometric functions.
  4. A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of exponential functions.
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