Harmonic Functions and What Analyticity Forces · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Harmonic functions from analytic ones

Mathematics · Complex Analysis · ages 20-21
Name ______________________   Date ____________

  1. Let u(x, y) equal x squared minus y squared. What is u_xx + u_yy at the point (3, 1)?

    Answer: ______________

  2. Let f(z) equal z squared. What is its real part u(x, y)?

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

  3. For f of z equal z squared, what is its real part u of x and y?

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

  4. If f equal u plus iv is analytic, then u_xx plus u_yy equals 0.

    Circle one:   True   False

  5. The pair u(x, y) equal x squared minus y squared and v(x, y) equal 2xy forms f(z) equal z squared. What is u(2, 1) plus v(1, 1)?

    Answer: ______________

  6. With u equal x squared minus y squared and v equal 2xy, what is u of (2, 1) plus v of (1, 1)?

    Answer: ______________

  7. The function u(x, y) equal 2xy is harmonic. Which v completes an analytic f equal u + iv, up to an added constant?

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

  8. The real and imaginary parts of 1 divided by z (for nonzero z) both satisfy Laplace equation, so 1 divided by z links complex analysis to 2D potential flow. True or false?

    Circle one:   True   False

  9. u equals x cubed minus 3x times y squared. Which is its harmonic conjugate, up to a constant?

    • x cubed plus y cubed
    • 3x squared y plus y cubed
    • 3x squared y minus y cubed

  10. Let u(x, y) equal x cubed minus 3x times y squared. Which is its harmonic conjugate, up to an added constant?

    • x cubed plus y cubed
    • 3x squared y minus y cubed
    • x cubed minus y cubed
    • 6xy
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Answer key

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Harmonic functions from analytic ones W1-mt_jqajP39XI4-s1

  1. 0 · u_xx equals 2 and u_yy equals minus 2, summing to 0.
  2. x squared minus y squared · Expanding (x + iy) squared gives real part x squared minus y squared.
  3. x squared minus y squared · Squaring x plus iy gives real part x squared minus y squared.
  4. True · Differentiating Cauchy-Riemann twice makes the mixed partials cancel.
  5. 5 · u(2, 1) equals 3 and v(1, 1) equals 2, totalling 5.
  6. 5 · u of (2, 1) is 3 and v of (1, 1) is 2, totalling 5.
  7. y squared minus x squared · y squared minus x squared satisfies both Cauchy-Riemann equations with u equal 2xy.
  8. True · True. Away from the origin both parts are harmonic, modelling a point source with its stream function.
  9. 3x squared y minus y cubed · This u is the real part of z cubed, whose imaginary part matches.
  10. 3x squared y minus y cubed · This u is the real part of z cubed, whose imaginary part is 3x squared y minus y cubed.
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