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 Goursat's theorem, which in addition to assuming the continuity of "f", also supposes 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 in Ω. Then "f" is holomorphic if and only if it satisfies the Cauchy-Riemann equation::: $frac{partial f}{partialar{z = frac{1}{2}left(frac{partial f}{partial x} + ifrac{partial f}{partial y}
ight)=0.$

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