The Argument Principle and Rouche's Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Count roots by watching winding

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

  1. A closed curve winds twice around the origin and never hits it. What is the winding number of the curve about the origin?

    Answer: ______________

  2. The argument principle says 1 divided by (2 pi i) times the loop integral of f prime over f equals what?

    • Zeros minus poles inside, counted with multiplicity
    • Zeros plus poles inside, counted with multiplicity
    • The degree of the polynomial f
    • Always zero for any loop

  3. One over 2 pi i times the loop integral of f prime over f equals what?

    • Zeros minus poles inside, counted with multiplicity
    • Zeros plus poles inside, counted with multiplicity
    • The degree of f on the whole plane

  4. In the argument principle, zeros and poles count with multiplicity.

    Circle one:   True   False

  5. On the circle of radius two, the leading term of a degree four polynomial dominates the rest and it vanishes four times inside. How many zeros does the full polynomial have inside?

    Answer: ______________

  6. A nonconstant analytic map sends open sets to what?

    • Every analytic map must be a polynomial
    • Zeros can pile up anywhere inside
    • Open sets must go to open sets

  7. On the unit circle, the term three z dominates z squared plus one in modulus. How many zeros does z squared plus three z plus one have strictly inside?

    • 0
    • 2
    • 1
    • 3

  8. If |f| is strictly larger than |g| on a closed contour, then f and f + g have the same zero count inside. True or false?

    Circle one:   True   False

  9. On your circle the two biggest terms tie in size. What follows?

    • That the dominant term still decides the count
    • That no count transfers and you must pick another circle
    • That the polynomial has no zeros at all

  10. A nonconstant analytic function maps open sets to open sets. Which statement follows?

    • Analytic functions map circles to circles
    • All zeros must lie on the domain boundary
    • The image of a disc is always a disc
    • No interior point can map to a maximum of |f|
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Answer key

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

Count roots by watching winding W1-mt_J1dC428M84-s1

  1. 2 · Two anticlockwise turns give winding number two.
  2. Zeros minus poles inside, counted with multiplicity · Each zero contributes plus its multiplicity and each pole minus its multiplicity.
  3. Zeros minus poles inside, counted with multiplicity · Each zero adds its multiplicity and each pole subtracts its own.
  4. True · A double zero winds twice and contributes two to the count.
  5. 4 · Rouche transfers the dominant term's headcount of four to the whole polynomial.
  6. Open sets must go to open sets · Winding rigidity forbids collapse, so images stay open and zeros stay isolated.
  7. 1 · The dominant term 3z has one zero inside, and Rouche transfers that count to the full polynomial.
  8. True · True. This is Rouche's theorem: dominance freezes the headcount.
  9. That no count transfers and you must pick another circle · Ties break the strict inequality Rouche needs, so no count transfers; pick another circle.
  10. No interior point can map to a maximum of |f| · A modulus peak would cap the image, contradicting openness around that value.
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