Let z = x + iy be a complex variable, and suppose that the function f : C → C given by f(z) = u(z) + iv(z) = u(x, y) + iv(x, y) is entire. Prove that h: C → R given by h(z) = h(x, y) = u^2(x, y) − v^2(x, y) is harmonic in C.

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.