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Analyticity of holomorphic functions

In complex analysis, a complex-valued functionf{\displaystyle f} of a complex variable z{\displaystyle z}: is said to be holomorphic at a point a{\displaystyle a} if it is differentiable at every point within some open disk centered at a{\displaystyle a}, andis said to be analytic at a{\displaystyle a} if in some open disk centered at a{\displaystyle a} it can be expanded as a convergentpower seriesf(z)=∑n=0∞cn(z−a)n{\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}} (this implies that the radius of convergence is positive).One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa.

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