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:

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

[edit] References

Narasimhan, Raghavan (2001), Complex Analysis in One Variable, Birkhäuser, ISBN 0817641645, <http://books.google.com/books?id=J-J4HmIDnOwC&pg=PA43&lpg=PA43&dq=%22Looman-Menchoff+theorem%22&source=web&ots=bBhleDsqtM&sig=Z2P6e4oBxpZDJeTqQAnXvsI6hr0&hl=en#PPA49,M1> .
Looman, H. (1923), “Über die Cauchy-Riemannschen Differeitalgleichungen”, Göttinger Nach.: 97-108 .
Menchoff, D. (1936), Les conditions de monogénéité, Paris .
Montel, P., “Sur les différentielles totales et les fonctions monogènes”, C. R. Acad. Sci. Paris 156: 1820-1822 .