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Everything about Analytic function

Type of function in mathematics

In mathematical analysis, an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at a point if, in some neighborhood of that point, it is equal to a power series centered there. The analytic function is therefore locally determined by coefficients of the series, or equivalently by derivatives of the function evaluated at the center of the series expansion. In other words, an analytic function is a function that is locally represented by a convergent Taylor series. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Analytic functionA 5-minute orientation
  2. Skim a recent paper: The characteristic function of Gaussian stochastic volatility models: an analytic expressionInfluential work since 2022
  3. Pick up a classic book: A course of modern analysisThe most-reprinted book on the subject
  4. Read the landmark paper: A high precision semi-analytic mass functionCited 41 times
  5. Go sideways: Power seriesA closely related topic
  6. Go sideways: Complex analysisA closely related topic

📖 Definitions

Noun Any smooth (infinitely differentiable) function f, defined on an open set D⊆ℂ (𝑜𝑟⊆ℝ), whose value in some neighbourhood of any given point x0∈D is given by the Taylor series ∑n=0∞f(n)(x0)n!(x−x0)n.

Noun window function

🗂️ The subject at a glance

  1. Definitions
  2. Examples
  3. Alternative characterizations
  4. Properties of analytic functions
  5. Analyticity and differentiability
  6. Real versus complex analytic functions
  7. Taylor series and radius of convergence
  8. Analytic continuation
  9. Analytic functions of several variables
  10. Analytic functions over other valued fields

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

🔗 Related topics