Complex Differentiability and the Cauchy-Riemann Equations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One derivative from every direction

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

  1. For u(x, y) = x squared - y squared, what is u_x at (3, 1)?

    Answer: ______________

  2. Which pair of equations is the Cauchy-Riemann system for u + iv?

    • u_x = v_y and u_y = -v_x
    • u_x = v_x and u_y = v_y
    • u_x = -v_y and u_y = v_x
    • u + v = 0

  3. Which pair of equations is the Cauchy-Riemann system for u plus iv?

    • u_x is v_y and u_y is negative v_x
    • u_x is v_x and u_y is v_y
    • u_x is negative v_y and u_y is v_x

  4. f(z) is the conjugate of z is differentiable nowhere. True or false?

    Circle one:   True   False

  5. For v(x, y) = 2xy, what is v_y at (3, 1)?

    Answer: ______________

  6. For u is x squared plus y squared with v = 0, where do the Cauchy-Riemann equations hold?

    • Everywhere
    • Only at the origin
    • Nowhere

  7. For f(z) = z squared, u_y at (1, 1) and -v_x at (1, 1) are what?

    • 2 and -2
    • both 2
    • 0 and 0
    • both -2

  8. Cauchy-Riemann plus continuous partials near z_0 guarantees differentiability at z_0. True or false?

    Circle one:   True   False

  9. Cauchy-Riemann plus continuous partials near z_0 guarantees differentiability at z_0. True or false?

    Circle one:   True   False

  10. f(z) = |z| squared is complex differentiable where?

    • everywhere
    • nowhere
    • only at 0
    • everywhere except 0
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Answer key

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

One derivative from every direction W1-mt_nZNmZcl9Ss-s1

  1. 6 · 6. Since u_x = 2x, at x = 3 it equals 6.
  2. u_x = v_y and u_y = -v_x · u_x = v_y and u_y = -v_x. Comparing real axis and imaginary axis approaches forces both.
  3. u_x is v_y and u_y is negative v_x · Comparing real axis and imaginary axis approaches forces both identities.
  4. True · Its u is x and v is negative y, giving u_x 1 against v_y negative 1 everywhere.
  5. 6 · 6. Since v_y = 2x, at x = 3 it equals 6, matching u_x there.
  6. Only at the origin · u_x is 2x meets v_y 0 only at x = 0, and u_y is 2y meets negative v_x 0 only at y = 0.
  7. both -2 · Both -2. u_y = -2x gives -2 and v_x = 2y gives 2, negated to -2.
  8. True · True. That continuity upgrades the equations from necessary to sufficient.
  9. True · That continuity upgrades the equations from necessary to sufficient.
  10. only at 0 · Only at 0. The equations hold solely at the origin, with continuous partials clinching differentiability there.
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