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→Real versus complex analytic functions: the given power series also diverges for |x| = 1 |
→Properties of analytic functions: Correction: paragraph describes the Identity Theorem, not the principle of permanence |
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* For any [[open set]] Ω ⊆ '''C''', the set ''A''(Ω) of all analytic functions ''u'' : Ω → '''C''' is a [[Fréchet space]] with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of [[Morera's theorem]]. The set <math>\scriptstyle A_\infty(\Omega)</math> of all [[bounded function|bounded]] analytic functions with the [[supremum norm]] is a [[Banach space]].
A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function ƒ has an [[accumulation point]] inside its [[___domain of a function|___domain]], then ƒ is zero everywhere on the [[connected space|connected component]] containing the accumulation point. In other words, if (''r<sub>n</sub>'') is a [[sequence]] of distinct numbers such that ƒ(''r''<sub>''n''</sub>) = 0 for all ''n'' and this sequence [[limit of a sequence|converges]] to a point ''r'' in the ___domain of ''D'', then ƒ is identically zero on the connected component of ''D'' containing ''r''. This is known as the [[
Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component.
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