Lemma of Osgood

The lemma Osgood , named after William Osgood is a statement from the theory of functions of several variables. A continuous function holomorphic in every variable is already holomorphic.

definition

Let it be an open set in the n -dimensional complex vector space . A function is called holomorphic in every variable if for all and all the functions ${\displaystyle U\subset \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle f\colon U\rightarrow \mathbb {C} }$ ${\displaystyle j\in \{1,\ldots ,n\}}$ ${\displaystyle a_{1},\ldots ,a_{j-1},a_{j+1},\ldots ,a_{n}\in \mathbb {C} }$

${\displaystyle \{z\in \mathbb {C} \mid (a_{1},\ldots ,a_{j-1},z,a_{j+1},\ldots ,a_{n})\in U\}\rightarrow \mathbb {C} ,\quad z\mapsto f(a_{1},\ldots ,a_{j-1},z,a_{j+1},\ldots ,a_{n})}$

are holomorphic , that is, if the functions resulting from the freezing of all but one variable are all holomorphic. ${\displaystyle f}$

statement

A holomorphic function is of course holomorphic in every variable. For the inverse, Osgood's lemma applies: ${\displaystyle f\colon U\rightarrow \mathbb {C} }$

• If there is an open set and a continuous map that is holomorphic in every variable, then is already holomorphic. ${\displaystyle U\subset \mathbb {C} ^{n}}$ ${\displaystyle f\colon U\rightarrow \mathbb {C} }$ ${\displaystyle f}$

comment

Because of the assumed continuity, one can iteratively apply and obtain the Cauchy integral formula for a closed poly-circle environment of a point ${\displaystyle {\overline {\Delta }}(a_{1},\ldots ,a_{n};r_{1}\ldots r_{n})\subset U}$ ${\displaystyle a=(a_{1},\ldots ,a_{n})\in U}$

${\displaystyle f(z_{1},\ldots ,z_{n})=\left({\frac {1}{2\pi \mathrm {i} }}\right)^{n}\int _{|a_{1}-\zeta _{1}|=r_{1}}\cdots \int _{|a_{n}-\zeta _{n}|=r_{n}}{\frac {f(\zeta _{1},\ldots ,\zeta _{n})}{(\zeta _{1}-z_{1})\cdots (\zeta _{n}-z_{n})}}\mathrm {d} \zeta _{1}\ldots \mathrm {d} \zeta _{n}}$     for     . ${\displaystyle (z_{1},\ldots ,z_{n})\in \Delta (a_{1},\ldots ,a_{n};r_{1}\ldots r_{n})}$

By developing the denominator of the integrand into a product of geometric series um , similar to the function theory of a variable, one obtains a power series expansion for um , which ends the proof. ${\displaystyle n}$ ${\displaystyle a_{j}}$ ${\displaystyle f}$ ${\displaystyle (a_{1},\ldots ,a_{n})}$

The statement from Osgood's lemma remains correct if the continuity requirement is dispensed with. This statement is then much more difficult to prove and is known as the Hartogs Theorem . For many applications, however, Osgood's lemma is sufficient, since the continuity is often clear.

Individual evidence

1. ^ William F. Osgood: Note on analytical functions of several variables , Mathematische Annalen 1899, Volume 52, pages 462-464
2. ^ Gunning - Rossi : Analytic functions of several complex variables . Prentice-Hall 1965, chap. I, Theorem 2 (Osgood's Lemma)
3. ^ Joseph L. Taylor: Several Complex Variables with Connections to Algebraic Geometry and Lie Groups , American Mathematical Society 2002, ISBN 0-8218-3178-X , Theorem 2.1.2