Complex Functions and the Complex Plane as a Domain · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Functions that remap the whole plane

Mathematics · Complex Analysis · ages 19-20
Name ______________________   Date ____________

  1. For w = z squared, u(x, y) = x squared - y squared. What is u(1, 2)?

    Answer: ______________

  2. For w = z squared with z = x + iy, what is the imaginary part v(x, y)?

    • x squared - y squared
    • x squared + y squared
    • 2xy
    • 0

  3. For w is z squared with z is x plus iy, what is the imaginary part v(x, y)?

    • x squared minus y squared
    • x squared plus y squared
    • 2xy

  4. Under w is z plus i, the real axis maps to the line Im(w) = 1. True or false?

    Circle one:   True   False

  5. For w = z squared, v(x, y) = 2xy. What is v(1, 2)?

    Answer: ______________

  6. For the limit of f(z) as z tends to z_0 to exist, f must do what?

    • Match along the real axis only
    • Approach the same value from every direction
    • Be defined at the target point

  7. Under w = 2z, the unit circle |z| = 1 maps to what?

    • the circle of radius 2
    • the circle of radius 1
    • a straight line
    • the circle of radius 4

  8. w = z squared sends the positive real axis to itself. True or false?

    Circle one:   True   False

  9. Under w is z squared, the point 1 plus i maps to what?

    • 2
    • 1 plus 2i
    • 2i

  10. Under w = z squared, the point 1 + i maps to what?

    • 2
    • -2i
    • 1 + 2i
    • 2i
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Answer key

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

Functions that remap the whole plane W1-mt_hMo851VTEs-s1

  1. -3 · -3. Note 1 - 4 = -3.
  2. 2xy · 2xy. Expanding (x + iy) squared gives real part x squared - y squared and imaginary part 2xy.
  3. 2xy · Expanding (x plus iy) squared gives real part x squared minus y squared and imaginary part 2xy.
  4. True · Adding i lifts every real point one unit up to imaginary part 1.
  5. 4 · 4. Note 2 times 1 times 2 = 4.
  6. Approach the same value from every direction · One rogue approach direction kills the limit, so all paths must agree.
  7. the circle of radius 2 · The circle of radius 2. Doubling stretches every radius 1 point to radius 2.
  8. True · True. Positive reals square to positive reals.
  9. 2i · Expanding (1 plus i) squared gives 1 plus 2i minus 1, which is 2i.
  10. 2i · 2i. Expanding (1 + i) squared gives 1 + 2i - 1 = 2i.
Worksheet · LightMySky