Hartogs Theorem (function theory)

As a set of Hartogs in will function theory of several complex variables , the basic message understood, after which with respect to each variable separately holomorphic function is holomorphic total. The sentence is named after the mathematician Friedrich Moritz Hartogs .

The lemma Osgood makes a similar statement, but it is assumed in this that the output function continuously is. This is therefore a special case of Hartogs' theorem.

statement

Be an open subset , be points, and be . For a function denote the function ${\displaystyle \Omega \subseteq \mathbb {C} ^{n}}$ ${\displaystyle a_{1},\dots a_{n}\in \mathbb {C} }$ ${\displaystyle \Omega _{j,a}:=\left\{z\in \mathbb {C} \,:\,(a_{1},\dots ,a_{j-1},z,a_{j+1},\dots ,a_{n})\in \Omega \right\}\subseteq \mathbb {C} }$ ${\displaystyle f\colon \Omega \rightarrow \mathbb {C} }$ ${\displaystyle f_{j,a}\colon \Omega _{j,a}\rightarrow \mathbb {C} }$

${\displaystyle z\mapsto f(a_{1},\dots ,a_{j-1},z,a_{j+1},\dots ,a_{n})}$.

If there is a holomorphic function for everyone and for everyone , then is holomorphic. ${\displaystyle f_{j,a}\colon \Omega _{j,a}\rightarrow \mathbb {C} }$ ${\displaystyle a_{1},\dots a_{n}\in \mathbb {C} }$ ${\displaystyle j\in \mathbb {N} :1\leq j\leq n}$ ${\displaystyle f\colon \Omega \rightarrow \mathbb {C} }$

interpretation

The sentence does not presuppose the continuity of the function , only the holomorphism with regard to the individual variables separately. By omitting the continuity condition, the proof becomes much more complicated, but it also shows clear differences to the real case: ${\displaystyle f}$

For example, the function does not have a continuous continuation in the point , but is real-analytic with respect to every variable. Hartogs' theorem excludes such a phenomenon for holomorphic functions. ${\displaystyle f\colon \mathbb {R} ^{2}\setminus \left\{(0,0)\right\}\rightarrow \mathbb {R} ,\;(x,y)\mapsto {\tfrac {x\cdot y}{x^{2}+y^{2}}}}$ ${\displaystyle (0,0)}$

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

literature

• Steven G. Krantz: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence, Rhode Island 1992.