Analytic function

A function that agrees locally with a convergent power series at every point in its domain. For complex functions, analyticity is equivalent to complex differentiability throughout an open neighborhood.

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Power series: An infinite series whose terms are powers of a variable multiplied by coefficients. Analytic functions are locally represented by convergent power series.
Cauchy–Riemann equations: Equations relating first partial derivatives that characterize complex differentiability under suitable regularity conditions. They express the local derivative constraints satisfied by complex analytic functions.
Real analytic function: A real-valued function locally equal to a convergent power series around every point of its domain. This is the real-variable counterpart, where smoothness does not imply analyticity.
Residue theorem: A theorem that evaluates contour integrals using the coefficients of singular terms at enclosed poles. Analyticity away from isolated singularities makes complex contour integrals computable.
Radius of convergence: The distance from a power series’ center within which the series converges. Each local power-series representation is valid only within its convergence region.
Cauchy's integral formula: A formula that expresses a holomorphic function’s interior values through its values on a surrounding contour. It yields derivatives of every order and power-series expansions from contour values.
Holomorphic function: A complex function that is complex differentiable at every point of an open domain. For complex functions on open domains, holomorphic and analytic describe equivalent conditions.
Generating function: A formal or convergent power series whose coefficients encode a sequence or combinatorial structure. Analytic properties of generating functions reveal coefficient growth and asymptotics.
Taylor series: A power series whose coefficients are determined by a function’s derivatives at a point. For analytic functions, the Taylor series recovers the function near its center.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an accumulation point agree throughout a connected domain. Local agreement determines the entire analytic function on a connected domain.