Laurent Series and Classifying Isolated Singularities · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Negative powers name the hole

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

  1. In 2/z² - 2/z² + 4/z + 5, how many nonzero negative powers remain after combining like terms?

    Answer: ______________

  2. The function 7/(z - (1 + i))^3 has a pole at which point?

    • 1 - i
    • -1 + i
    • 1 + i

  3. The function 1/z has which kind of singularity at the origin?

    • A removable singularity
    • A pole of order 1
    • An essential singularity

  4. A function with a removable singularity can be redefined at that single point to become analytic there.

    Circle one:   True   False

  5. What is the coefficient of 1/z in the Laurent expansion of 12z² e^(1/z) about 0?

    Answer: ______________

  6. Near a pole, how does the function behave?

    • |f(z)| tends to infinity along every approach
    • Values approach positive real infinity along every approach
    • The function stays bounded near the point

  7. On 1 < |z| < 3, which series equals 1/(z - 1) + 1/(z - 3)?

    • Sum over n >= 0 of z^(-n-1) minus sum over n >= 0 of z^n/3^(n+1)
    • Sum over n >= 0 of z^n minus sum over n >= 0 of z^(-n-1)/3^(n+1)
    • Sum over n >= 0 of z^n/3^(n+1) minus sum over n >= 0 of z^(-n-1)

  8. Near an essential singularity, density of values forces e^(1/z) to take the value 0. True or false?

    Circle one:   True   False

  9. On a punctured disk about a, the function ((z - a)^2 + 1)/(z - a)^5 - 1/(z - a)^5 + 2/(z - a) has which classification?

    • Pole of order 5, reading the deepest written power
    • Pole of order 3, after cancelling the order 5 terms
    • Pole of order 2, counting the two remaining negative terms

  10. The function (z - 1)/((z - 1)(z - 3)) is undefined at 1 and 3. Its Laurent expansion about 1 on |z - 1| > 2 has infinitely many negative powers. Classify both singularities.

    • 1 removable with filled value -1/2; 3 a simple pole
    • 1 essential from the outer annulus; 3 a simple pole
    • 1 a simple pole; 3 removable with filled value -1/2
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Answer key

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

Negative powers name the hole W1-mt_WoQV0gD1O--s1

  1. 1 · The z² terms cancel, leaving 4/z + 5 with one nonzero negative power.
  2. 1 + i · The negative power is (z - (1 + i))^(-3), so the centre is 1 + i.
  3. A pole of order 1 · For 1/z the exponent is -1, so the order is 1.
  4. True · Holomorphic and bounded on a punctured disk gives a finite limit that fills the point.
  5. 2 · The term with n = 3 gives exponent -1, and 12/3! = 2.
  6. |f(z)| tends to infinity along every approach · Uniform escape to infinity is the pole signature, never dense wandering.
  7. Sum over n >= 0 of z^(-n-1) minus sum over n >= 0 of z^n/3^(n+1) · On 1 < |z| < 3, 1/(z - 1) uses ratio 1/z and 1/(z - 3) uses ratio z/3.
  8. False · Density means values land arbitrarily close to 0, not that 0 is attained; e^w is never 0.
  9. Pole of order 3, after cancelling the order 5 terms · Cancelling the order 5 terms leaves 1/(z - a)^3 + 2/(z - a), so the order is 3.
  10. 1 removable with filled value -1/2; 3 a simple pole · Away from 1, f(z) = 1/(z - 3), so 1 is removable with limit -1/2 and 3 is a simple pole.
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