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. In the Laurent expansion 2/z - 2/z + 5/z + 3, what is the residue at the origin?

    Answer: ______________

  2. What is the order of the pole of 1/z^3 at the origin, and what is its residue?

    • Order 3, residue 0
    • Order 3, residue 1
    • Order 1, residue 0

  3. What is the residue at the origin of 1/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 f(z) = z^2/(2z - 4), what is the residue at the simple pole z=2?

    Answer: ______________

  6. A function f is holomorphic on a simply connected domain except isolated singularities at z=1 and z=2. A closed contour winds twice counterclockwise around z=1 where Res(f,1) = 3, and once clockwise around z=2 where Res(f,2) = -1. There are no other singularities. What is the integral?

    • 2 pi i times 5
    • 2 pi i times 6
    • 2 pi i times 7

  7. A contour passes directly through a simple pole of f. What can we say about the ordinary contour integral?

    • It equals half the residue times 2 pi i
    • It is not defined as an ordinary contour integral
    • It equals the residue times 2 pi i

  8. The higher-pole derivative formula can be used to find the residue of e^(1/z) at 0.

    Circle one:   True   False

  9. Let f(z) = (2z^3 + 1)/(z^3 (z-1)), holomorphic on a simply connected domain except at 0 and 1. A closed piecewise C1 contour winds once counterclockwise around 0 and twice counterclockwise around 1. What is the integral?

    • 2 pi i times 3
    • 2 pi i times 5
    • 2 pi i times 10

  10. Consider f(z) = 1/(z(z-1)(z-3)) and a contour that winds once counterclockwise around z=0 and z=1, and once clockwise around z=3. What is the integral?

    • -2 pi i / 3
    • 2 pi i / 3
    • pi i / 3
LightMySky · lightmysky.comW1-mt_BN2biZ_zos-s1

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. 5 · 2/z - 2/z cancels, leaving 5/z + 3 with residue 5.
  2. Order 3, residue 0 · The Laurent expansion is z^(-3), so order 3 and no z^(-1) term.
  3. 1 · The Laurent expansion is the single term 1/z, so the coefficient is 1.
  4. True · All other powers have antiderivatives, whose loop integrals vanish.
  5. 2 · g(2)/h'(2) = 4/2 = 2.
  6. 2 pi i times 7 · Twice 3 plus once -1 times -1 is 7, scaled by 2 pi i.
  7. It is not defined as an ordinary contour integral · The residue theorem requires no singularities on the contour.
  8. False · e^(1/z) has an essential singularity, not a finite-order pole.
  9. 2 pi i times 5 · Triple pole residue -1 at 0 and simple pole residue 3 at 1 with windings +1 and +2 give 5, times 2 pi i.
  10. -2 pi i / 3 · Residues 1/3, -1/2, 1/6 with windings +1, +1, -1 give -1/3, times 2 pi i.
Worksheet · LightMySky