Hartogs' theorem

As a set of Hartogs is understood in the theory of functions of several complex variables, the basic statement that one with respect to each variable separately holomorphic function is holomorphic in total.

Preparation

Is an open subset, are points. Then that subset of call in which all the coordinates other than the -th corresponding points. For a function call the function.

Statement

Be an open subset, a function that applies to that: is a holomorphic function. Then is holomorphic.

Interpretation

In the sentence, the continuity of the function is not required, only the holomorphy separately with respect to each variable. By omitting the steadiness condition of the evidence is substantially complicated, but also shows significant differences from the real case:

For example, the function has no continuous extension of the point, but is real - analytic with respect to each variable. The set of Hartogs excludes such a phenomenon for holomorphic function.

From the standpoint of partial differential equations, the set of Hartogs can also be interpreted such that the solutions of the Cauchy -Riemann differential equations in real differentiability without further regularity assumptions with respect to all variables are automatically holomorphic.