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 be 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. This shows that the function f must be assumed continuous in the theorem.

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). This shows that a naive generalization of the Looman–Menchoff theorem to a single point is false:

Let f be continuous at a neighborhood of a point z, and such that $\partial f/\partial x$ and $\partial f/\partial y$ exist at z. Then f is holomorphic at z if and only if it satisfies the Cauchy–Riemann equation at z.

References

Gray, J. D.; Morris, S. A. (1978), "When is a Function that Satisfies the Cauchy-Riemann Equations Analytic?", The American Mathematical Monthly (published April 1978), 85 (4): 246–256, doi:10.2307/2321164, JSTOR 2321164.
Looman, H. (1923), "Über die Cauchy–Riemannschen Differentialgleichungen", Göttinger Nachrichten: 97–108.
Menchoff, D. (1936), Les conditions de monogénéité, Paris{{citation}}: CS1 maint: location missing publisher (link).
Montel, P. (1913), "Sur les différentielles totales et les fonctions monogènes", C. R. Acad. Sci. Paris, 156: 1820–1822.
Narasimhan, Raghavan (2001), Complex Analysis in One Variable, Birkhäuser, ISBN 0-8176-4164-5.

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