Multiple choice

For an analytic function f(x + iy) = u(x, y) + i v(x, y), u is given by u = 3x2 – 3y2. The expression for v, considering K to be a constant is

  1. 3y2 – 3x2 + K

  2. 6x – 6y + K

  3. 6y – 6x + K

  4. 6xy + K
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D Correct answer
Explanation

For an analytic function, u and v satisfy the Cauchy-Riemann equations: du/dx = dv/dy and du/dy = -dv/dx. Given u = 3x^2 - 3y^2, du/dx = 6x and du/dy = -6y. Thus, dv/dy = 6x (so v = 6xy + f(x)) and dv/dx = 6y. Differentiating v = 6xy + f(x) with respect to x gives 6y + f'(x) = 6y, so f'(x) = 0. Thus, v = 6xy + K.

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