mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Oka's lemma, proved by

Kiyoshi Oka was a Japanese mathematician who did fundamental work in the theory of several complex variables. Biography Oka was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924. He was in ...

, states that in a

domain of holomorphy In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. Forma ...

\Complex^n

, the function

-\log d(z)

is plurisubharmonic, where

d

is the distance to the boundary. This property shows that the domain is pseudoconvex. Historically, this lemma was first shown in the Hartogs domain in the case of two variables. Furthermore, Oka's lemma is the inverse of Levi's problem (unramified Riemann domain over

\Complex^n

). Perhaps, this is why Oka referred to Levi's problem as "problème inverse de Hartogs", and could explain why Levi's problem is occasionally referred to as Hartogs' Inverse Problem.

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