Casorati–Weierstrass theorem

The Casorati–Weierstrass theorem states that near an isolated essential singularity, a holomorphic function takes values dense in the complex plane in every punctured neighborhood.

Essential singularity: An isolated singularity of a holomorphic function that is neither removable nor a pole. The theorem’s hypothesis is precisely that the singularity is essential.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphy supplies the analytic structure required for the singularity classification.
e^(1/z): The function defined by e^(1/z) on the complex plane with zero removed. Its essential singularity at zero illustrates dense local values through exponential periodicity.
Karl Theodor Wilhelm Weierstrass: A German mathematician whose work shaped nineteenth-century analysis and the foundations of function theory. The theorem bears Weierstrass’s name and belongs to the analytic theory he developed.
Removable singularity: An isolated singularity at which a holomorphic function extends holomorphically after assigning a value. A removable singularity has controlled local behavior, unlike the theorem’s essential case.
Isolated singularity: A point where a function is undefined but holomorphic throughout some surrounding punctured disk. The theorem concerns the local behavior around this specific kind of exceptional point.
sin(1/z): The function defined by sin(1/z) on the complex plane with zero removed. Its essential singularity at zero produces values approaching every complex number.
Felice Casorati: An Italian mathematician known for work in complex analysis and algebra. Casorati’s name is attached to the density theorem for essential singularities.
Pole (complex analysis): An isolated singularity where a holomorphic function tends to infinity, with a finite-order principal part. A pole also lacks dense local values; near it, the function instead diverges.
Maximum modulus principle: A theorem stating that a nonconstant holomorphic function cannot attain a local maximum of its modulus inside its domain. It underlies the boundedness reasoning used to distinguish essential behavior.

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Émile PicardCompared with: Picard’s theorem strengthens this density conclusion to nearly all values, attained infinitely often.