Analyticity of holomorphic functions

Every function holomorphic on an open subset of the complex plane equals a convergent power series in a neighborhood of each point. Its Taylor coefficients are determined by derivatives, and its radius of convergence reaches at least the nearest obstruction to holomorphic continuation.

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Cauchy's integral formula: A formula expressing a holomorphic function inside a contour through its values along that contour. Its kernel expands as a geometric series, yielding the local power series and coefficient bounds.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphicity is the exact hypothesis of the local power-series theorem.
Smooth function: A function with continuous derivatives of every finite order. Smoothness permits Taylor coefficients but does not ensure their series recovers the function.
Isolated zeros theorem: A theorem stating that a nonzero holomorphic function has isolated zeros in its domain. A first nonzero Taylor coefficient makes each zero locally a finite-order factor times a nonvanishing function.
Taylor series: A power series whose coefficients are the derivatives of a function at a chosen point. Holomorphic functions equal their Taylor series locally, not merely share its formal coefficients.
Complex derivative: The limit defining the instantaneous rate of change of a complex function with respect to a complex variable. A single complex derivative requires compatible behavior from every direction in the plane.
Flat function: A smooth function whose derivatives of every order vanish at a point while the function is not locally zero. The standard real-variable example shows why infinite differentiability is weaker than analyticity.
Analytic continuation: The extension of an analytic function beyond its initial domain while preserving its local values. Overlapping local power series can extend a function beyond one expansion disk.
Cauchy estimates: Bounds on derivatives of holomorphic functions obtained from Cauchy's integral formula. These bounds force Taylor coefficients to decay fast enough for convergence near the expansion point.
Open set: A subset of a topological space in which every point has a neighborhood contained in the set. The theorem expands around points with surrounding room inside the function's domain.