Looman–Menchoff theorem

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:

• Let Ω be an open set in C and f : Ω → C a continuous function. Suppose that the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$ exist everywhere but a countable set in Ω. Then f is holomorphic if and only if it satisfies the Cauchy–Riemann equation:

$\frac{\partial f}{\partial\bar{z}} = \frac{1}{2}\left(\frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right)=0.$

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).

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