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Let f(z) be analytic in some domain. Show that f(z) is necessarily a constant if

ID: 2973540 • Letter: L

Question

Let f(z) be analytic in some domain. Show that f(z) is necessarily a constant if either the conjugate of f(z) is analytic or f(z) assumes only pure imaginary values in the domain.

Explanation / Answer

By hypothesis, (with f = u + iv and z = x = iy) both f(z) = u(x,y) + i v(x, y) and conj(f(z)) = u(x,y) - i v(x, y) are analytic. So, applying the Cauchy-Riemann Equations to each equation, we get f(z) = u(x,y) + i v(x, y) ==> ?u/?x = ?v/?y and ?u/?y = -?v/?x. conj(f(z)) = u(x,y) - i v(x, y) ==> ?u/?x = -?v/?y and ?u/?y = ?v/?x. Adding corresponding pairs of equations together, we get ?u/?x = 0 and ?u/?y = 0 ==> u is constant. Subtracting corresponding pairs of equations together, we get ?v/?x = 0 and ?v/?y = 0 ==> v is constant. Hence, f is constant.

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