Holomorphic function

A complex-valued function is holomorphic on an open set when it has a complex derivative at every point of that set. This condition forces strong local structure, including power-series expansions.

Complex derivative: The limit of a difference quotient for a complex-valued function as its complex increment approaches zero. Its existence at every point of an open domain defines holomorphicity.
Complex plane: The plane whose points represent complex numbers using real and imaginary coordinates. Holomorphic functions are studied on open subsets of this plane.
Contour integration: The integration of complex-valued functions along oriented curves in the complex plane. Holomorphicity enables powerful integral theorems for contours and their interiors.
Real differentiability: Differentiability of a function between real vector spaces, defined by a linear approximation with an error small relative to the increment. Unlike real differentiability, complex differentiability requires one derivative limit across all directions.
Cauchy–Riemann equations: Partial-derivative equations that characterize complex differentiability when the relevant derivatives are continuous. They translate the complex derivative condition into relations between real and imaginary parts.
Open set: A set in a topological space in which every point has a neighborhood contained in the set. The domain must be open for holomorphicity to describe behavior around each point.
Conformal mapping: A mapping that preserves angles locally wherever its derivative is nonzero. Holomorphic functions with nonzero derivative generate conformal maps.
Meromorphic function: A function holomorphic except at isolated poles, where its magnitude diverges. It extends holomorphic functions by permitting isolated pole singularities.
Power series: An infinite sum of powers of a variable, with coefficients that determine where the series converges. Locally, every holomorphic function equals a convergent power series.
Complex differentiability: Differentiability defined by a complex difference quotient having the same limit from every direction of approach. It is the pointwise condition that holomorphicity requires throughout an open domain.

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