Math 2.0

Real functions Differentiation Definition

Analytic function

Builds on Power series

Definition

Let II be an open interval and let f ⁣:I→ℝf \colon I \to \mathbb{R}. The function ff is analytic at a∈Ia \in I if there are r>0r > 0 and a power series ∑cn(x−a)n\sum c_n (x - a)^n that converges to f(x)f(x) for all xx with ∣x−a∣<r|x - a| < r. It is analytic on II if it is analytic at each point of II.

Remarks

An analytic function is smooth, and its power series at aa is its Taylor series at aa. Polynomials, the exponential function, sine, and cosine are analytic on ℝ\mathbb{R}. A smooth function need not be analytic. See the smooth non-analytic function. Mathlib defines analyticity at a point as AnalyticAt.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionPower series, definition, no evidence levelPower seriesdefinitionAnalytic function, definition, no evidence levelAnalytic functiondefinition
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  1. Sequence definition
  2. Real sequence definition
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  5. Analytic function definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Analytic_function, Definitions
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Aliasesreal-analytic-function
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WritingW1 stated
MSC26E05
DefinesAnalytic function
UsesPower series

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