Hartogs extension theorem

In complex dimension at least two, holomorphic functions defined outside suitable compact sets often extend holomorphically across them. The phenomenon has no general one-variable counterpart.

Hartogs figure: A union of two overlapping polydiscs arranged to expose a hole in several complex variables. It provides a local geometric model in which holomorphic data force extension into a missing region.
Removable singularity: An isolated point where a function is undefined but admits a holomorphic extension. Hartogs extension generalizes removability from isolated points to certain compact holes in higher dimensions.
Friedrich Hartogs: A German mathematician whose work established foundational results in several complex variables. His name is attached to the extension phenomenon he proved.
Hartogs extension theorem for several complex variables: A local extension result asserting that holomorphic functions cross certain compact holes in domains of complex dimension at least two. This formulation makes explicit the local setting and dimension condition behind the theorem.
Cauchy integral formula: An integral identity expressing a holomorphic function's values through its values on a surrounding contour. Integral representations help construct extensions from boundary values in several complex variables.
Laurent series: A complex power series that allows both positive and negative integer powers. In one variable, negative powers describe isolated singularities that obstruct extension.
Hartogs's 1906 theorem: Hartogs's early result showing that holomorphic functions in several variables can extend across certain missing regions. It is the historical formulation underlying the modern extension theorem.
Hartogs phenomenon: The unexpected extension of holomorphic functions across holes in several complex variables. This name captures the broader behavior exhibited by the theorem.
Several complex variables: The study of holomorphic functions and geometry in complex spaces of dimension greater than one. The theorem depends on phenomena unique to complex dimension greater than one.
Essential singularity: An isolated singularity whose Laurent series has infinitely many negative-power terms. Such singularities show why arbitrary one-variable punctures do not admit holomorphic extension.

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