In the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations. It is thus a generalization of a theorem by Édouard Goursat, which instead of assuming the continuity of f, assumes its Fréchet differentiability when regarded as a function from a subset of R2 to R2.
A complete statement of the theorem is as follows:
Examples
Looman pointed out that the function given by f(z) = exp(−z−4) for z ≠ 0, f(0) = 0 satisfies the Cauchy–Riemann equations everywhere but is not analytic, or even continuous, at z = 0.
The function given by f(z) = z5/|z|4 for z ≠ 0, f(0) = 0 is continuous everywhere and satisfies the Cauchy–Riemann equations at z = 0, but is not analytic at z = 0 (or anywhere else).