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. Consider one divided by z plus one divided by z squared plus z, valid for nonzero z. How many negative powers of z appear in this Laurent expansion?

    Answer: ______________

  2. The function 1 divided by z has which kind of singularity at the origin?

    • A removable singularity
    • A pole of order 1
    • A pole of order 2
    • An essential singularity

  3. The function 1 divided by 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. For one divided by z squared plus three divided by z plus z, what is the coefficient of one divided by z?

    Answer: ______________

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

    • Poles head to infinity from every side
    • Poles take values dense near every number
    • Poles stay bounded near the point

  7. On the annulus 0 below |z| with no upper bound, which series equals e to the power one divided by z?

    • Sum of z^-n over n factorial
    • The sum of z to the power n divided by n factorial
    • Just 1 divided by z
    • A finite polynomial in 1 divided by z

  8. Near an essential singularity, the function stays large in absolute value, just like near a pole. True or false?

    Circle one:   True   False

  9. A Laurent expansion shows negative powers running 1, 2, 3, without end. What is the singularity?

    • It must be a pole of order 2
    • It must be essential
    • It must be removable

  10. Which behavior separates an essential singularity from a pole?

    • The function is unbounded near the point
    • The Laurent series has finitely many negative powers
    • In every punctured neighbourhood the image is dense
    • The limit at the point exists and is infinite
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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. 2 · The terms one divided by z and one divided by z squared give two negative powers.
  2. A pole of order 1 · A single 1 divided by z term is a pole of order 1.
  3. A pole of order 1 · One lone 1 over z term is a pole, and depth 1 sets the order.
  4. True · Assigning the finite limit as the value completes an analytic function.
  5. 3 · The one divided by z term carries coefficient three.
  6. Poles head to infinity from every side · Uniform escape to infinity is the pole signature, never dense wandering.
  7. Sum of z^-n over n factorial · Substituting 1 divided by z into the exponential series gives negative powers running to infinity.
  8. False · False. Essential singularities hit values near everything, including near zero, as e to one divided by z shows.
  9. It must be essential · Infinitely many negative powers is exactly the essential fingerprint.
  10. In every punctured neighbourhood the image is dense · Density of values is the Casorati mark of essential singularities. Poles simply escape to infinity.
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