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

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

  1. At (1, 2), u_x = 5 and CR holds. What is v_y?

    Answer: ______________

  2. When deriving CR, the imaginary-axis quotient (u_y + i v_y) / i simplifies to what?

    • v_y - i u_y
    • u_y - i v_y
    • -v_y + i u_y

  3. 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

  4. f(z) = conj(z) is differentiable nowhere. True or false?

    Circle one:   True   False

  5. For f(z) = z², what is the imaginary part of f'(z) at 1 + 2i?

    Answer: ______________

  6. For u = x² + y² with v = 0, where do the Cauchy-Riemann equations hold?

    • Everywhere
    • Nowhere
    • Only at 0

  7. For u = 2x + 3y, find p and q in v = px + qy so CR holds everywhere.

    • p = 3, q = 2
    • p = -3, q = 2
    • p = -3, q = -2

  8. A student finds that partials are not continuous at z0 and concludes f is not differentiable there. This inference is valid. True or false?

    Circle one:   True   False

  9. At z0, u_x = 4, v_y = 4, u_y = -3, v_x = 3, with all four partials continuous near z0. Then f'(z0) = 4 - 3i. True or false?

    Circle one:   True   False

  10. For f(z) = y² with z = x + iy, classify the differentiability set and holomorphic neighbourhoods.

    • Everywhere, holomorphic everywhere
    • Real axis only, nowhere holomorphic
    • Nowhere, nowhere holomorphic
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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. 5 · CR gives v_y = u_x = 5 at the point.
  2. v_y - i u_y · Dividing by i rotates the vector: (u_y + i v_y)/i = v_y - i u_y.
  3. u_x = v_y and u_y = -v_x · Comparing real-axis and imaginary-axis approaches forces both identities.
  4. True · Its u = x and v = -y give u_x = 1 against v_y = -1 everywhere.
  5. 4 · z² has continuous polynomial partials satisfying CR everywhere, so f'(z) = u_x + i v_x = 2z = 2 + 4i, imaginary part 4.
  6. Only at 0 · u_x = 2x meets v_y = 0 only at x = 0, and u_y = 2y meets -v_x = 0 only at y = 0.
  7. p = -3, q = 2 · u_x = 2 = v_y = q, and u_y = 3 = -v_x = -p, so p = -3 and q = 2.
  8. False · False. The sufficient test is unavailable, but a direct quotient can still settle the derivative.
  9. False · CR holds and partials are continuous, so f'(z0) = u_x + i v_x = 4 + 3i, not 4 - 3i.
  10. Real axis only, nowhere holomorphic · CR holds only when y = 0, giving differentiability on the real axis, but no open set of differentiability.
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