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An analytic function of a complex variable z = x + i y is expressed as f(z) = u(x, y) + i v(x, y), where i = √−1. If u(x, y) = x² − y², then expression for v(x, y) in terms of x, y and a general constant c would be

1. xy + c
2. (x² + y²)/2 + c
3. 2xy + c
4. ((x − y)²)/2 + c

Correct answer: 2xy + c

Solution

With u = x^2 - y^2, u_x = 2x = v_y gives v = 2xy + g(x), and u_y = -2y = -v_x forces g'(x)=0. Hence v = 2xy + c (index 2), not xy + c.

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