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 [math]\displaystyle{ \partial f/\partial x }[/math] and [math]\displaystyle{ \partial f/\partial y }[/math] exist everywhere but a countable set in Ω. Then f is holomorphic if and only if it satisfies the Cauchy–Riemann equation:

[math]\displaystyle{ \frac{\partial f}{\partial\bar{z}} = \frac{1}{2}\left(\frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right)=0. }[/math]

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 [math]\displaystyle{ \partial f/\partial x }[/math] and [math]\displaystyle{ \partial f/\partial y }[/math] 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 85 (4): 246–256, April 1978, doi:10.2307/2321164 .
• Looman, H. (1923), "Über die Cauchy–Riemannschen Differentialgleichungen", Göttinger Nachrichten: 97–108 .
• Menchoff, D. (1936), Les conditions de monogénéité, Paris .
• 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, https://books.google.com/books?id=J-J4HmIDnOwC&q=%22Looman-Menchoff+theorem%22&pg=PA43 .