Conformal Maps and Mobius Transformations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Keep angles and bend regions

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

  1. For f(z)=iz, what is the real part of f(2+5i)?

    Answer: ______________

  2. For f(z)=2iz, how do tangent lengths and directions change?

    • Lengths double; directions turn 90 degrees clockwise
    • Lengths double; directions turn 90 degrees counterclockwise
    • Lengths halve; directions turn 90 degrees counterclockwise
    • Lengths stay fixed; directions turn 180 degrees counterclockwise

  3. Why is f(z)=e^z conformal at every finite complex point?

    • Its derivative is zero at every complex point
    • Its derivative e^z is nonzero at every point
    • It sends every straight line to a straight line

  4. Given two ordered triples on the extended complex plane, each with three distinct entries, exactly one Mobius map sends each source entry to its corresponding target entry.

    Circle one:   True   False

  5. A Mobius map T sends 0 to 1, 1 to 0 and infinity to infinity. Find T(5).

    Answer: ______________

  6. Which listed map is a bijection from H={Im z>0} onto D={|w|<1} and sends i to 0?

    • C(z)=(z+i)/(z-i)
    • C(z)=z/(z+1)
    • C(z)=(z-i)/(z+i)

  7. For T(z)=(2z+1)/(z+2), what is T'(0)?

    • 3/4
    • 1/2
    • 1/4
    • 3/2

  8. Because e^z is conformal at every finite point, it is one-to-one on the whole complex plane.

    Circle one:   True   False

  9. Treat the real line together with infinity. What kind of set is its image under a Mobius map?

    • It is always a straight line through the origin
    • It is a generalized circle: a circle or a line
    • It is always a polygon with finitely many sides

  10. Let C(z)=(z-i)/(z+i) and u(w)=Re(w^2) on the unit disc. For the harmonic pullback v(z)=u(C(z)), find v(3i).

    • -1/4
    • 1/2
    • -1/2
    • 1/4
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Answer key

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

Keep angles and bend regions W1-mt_WtHCbAT4GI-s1

  1. -5 · i(2+5i)=2i+5i^2=-5+2i. The real part is -5.
  2. Lengths double; directions turn 90 degrees counterclockwise · The multiplier 2i has modulus 2 and argument pi/2. It doubles tangent lengths and turns directions 90 degrees counterclockwise.
  3. Its derivative e^z is nonzero at every point · The exponential is holomorphic and has derivative e^z, which never vanishes. The local conformality criterion applies everywhere.
  4. True · True. Normalize the source and target triples in matching order with A and B. Then B^-1 composed with A realizes the prescribed correspondence, and the three-point theorem gives uniqueness.
  5. -4 · Fixing infinity makes T affine. The other conditions give T(z)=1-z, so T(5)=-4.
  6. C(z)=(z-i)/(z+i) · C(z)=(z-i)/(z+i) sends i to 0, maps H into D, and has inverse i(1+w)/(1-w) taking all of D into H.
  7. 3/4 · ad-bc=2*2-1*1=3. Therefore T'(z)=3/(z+2)^2 and T'(0)=3/4.
  8. False · Local conformality does not imply global injectivity. The distinct inputs 0 and 2*pi*i both map to 1.
  9. It is a generalized circle: a circle or a line · The completed real line is a generalized circle. Mobius maps preserve that family, consisting of circles and completed lines.
  10. 1/4 · C(3i)=2i/(4i)=1/2. Then u(1/2)=Re((1/2)^2)=1/4.
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