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:
::$\frac = \frac\left(\frac + i\frac\right)=0.$

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'') = ''z''⁵/, ''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

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Theorems in complex analysis

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