The Residue Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One coefficient rules each loop

Mathematics · Complex Analysis · ages 21-22
Name ______________________   Date ____________

  1. What is the residue of seven divided by z at zero?

    Answer: ______________

  2. What is the residue at the origin of one divided by z?

    • 1
    • 0
    • minus 1
    • 2 pi i

  3. What is the residue at the origin of one divided by z?

    • 1
    • 0
    • 2 pi i

  4. Term by term loop integration kills every Laurent power except minus one.

    Circle one:   True   False

  5. For (z plus three) divided by (z minus one), the residue at the simple pole equals the numerator evaluated there. That value is three plus one. What is it?

    Answer: ______________

  6. A loop winds twice where residue is 3 and once where residue is minus 1. What is the integral?

    • 2 times 2 pi i
    • 5 times 2 pi i
    • 4 times 2 pi i

  7. What is the residue of 1 divided by z squared at the origin?

    • 1
    • 2
    • 0
    • minus 1

  8. The residue theorem multiplies each enclosed residue by the winding number of the contour about that pole. True or false?

    Circle one:   True   False

  9. You traverse your contour clockwise instead of anticlockwise. What changes?

    • Clockwise traversal flips every sign
    • Clockwise traversal doubles each residue
    • Direction never matters at all

  10. What is the integral of (2z + 1) divided by (z times (z minus 3)) around the circle |z| equal 2?

    • 0
    • 2 pi i
    • 2 pi i times (one third)
    • 2 pi i times (minus one third)
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Answer key

For grown-ups. Fold this page away before handing over the rest.

One coefficient rules each loop W1-mt_BN2biZ_zos-s1

  1. 7 · The one divided by z coefficient is seven.
  2. 1 · The Laurent expansion is a single one divided by z term with coefficient 1.
  3. 1 · The function already is one 1 over z term with coefficient 1.
  4. True · Every other power owns an antiderivative, whose closed loop integral is zero.
  5. 4 · Evaluating z plus three at one gives four.
  6. 5 times 2 pi i · Twice 3 plus once minus 1 is 5, scaled by 2 pi i.
  7. 0 · There is no one divided by z term in the expansion, so the residue is 0.
  8. True · True. That weighting is exactly the general statement of the theorem.
  9. Clockwise traversal flips every sign · Clockwise means negative winding, so each contribution changes sign.
  10. 2 pi i times (minus one third) · Only the pole at 0 is inside, with residue (2 times 0 + 1) divided by (0 minus 3), which is minus one third. Times 2 pi i.
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