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Calculus with complex numbers: residues and contour integrals.

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Opens at level 60.

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Builds on: Complex numbers (not open yet) · Series (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Roots of unity par 20 s not tried yet

  worked example

  What is the sum of the 12th powers of all 12th roots of unity?

  Answer: 12

  1. Each of the 12 roots ω has ω¹² = 1.
  2. So the sum is 12 × 1 = 12.

• Residues par 30 s not tried yet

  worked example

  What is the residue of f(z) = 1/((z − 5)(z + 4)) at z = −4? Give an exact number or fraction.

  Answer: -1/9

  1. z = −4 is a simple pole: f(z) = g(z)/(z + 4) with g(z) = 1/(z − 5).
  2. The residue is g(−4) = 1/(−4 − 5) = −1/9.

• Contour integrals par 40 s not tried yet

  worked example

  The integral of f(z) = (z² + 3z + 4)/z around the circle |z| = 3 (counterclockwise) equals kπi. What is k?

  Answer: 8

  1. The only pole, z = 0, lies inside |z| = 3.
  2. Its residue is the numerator at z = 0, which is its constant term: 4.
  3. Residue theorem: 2πi × 4 = 8πi, so k = 8.

• Analytic or not? par 30 s not tried yet

  worked example

  Three of these functions of z = x + iy are analytic (complex-differentiable) everywhere. Which one is not?

  1. conj(z)
  2. z²
  3. z² + 3z
  4. z³

  Answer: conj(z)

  1. conj(z) = x − iy: ∂u/∂x = 1 but ∂v/∂y = −1, so Cauchy–Riemann fails everywhere.
  2. The other three, z³, z² + 3z and z², are analytic everywhere: polynomials in z, or built from eᶻ.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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