Math 2.0

Ordinary differential equations Definition

Resurgent function

Builds on Holomorphic functionAnalytic continuationBorel summation

Definition

Let Ω\Omega be a closed discrete subset of ℂ\C. A power series φ^(ζ)=∑n≥0cnζn\hat\varphi(\zeta) = \sum_{n \ge 0} c_n \zeta^n with positive radius of convergence is Ω\Omega-continuable if it satisfies both conditions:

  1. Its sum is a holomorphic function on a disc DD centred at 00 with D∩Ω⊆{0}D \cap \Omega \subseteq \set{0}.
  2. It has an analytic continuation along every path in ℂ∖Ω\C \setminus \Omega that starts in D∖{0}D \setminus \set{0}.

A formal series φ~(z)=∑n≥0anz−n−1\tilde\varphi(z) = \sum_{n \ge 0} a_n z^{-n-1} in the variable zz near ∞\infty is an Ω\Omega-resurgent series if its formal Borel transform φ^(ζ)=∑n≥0ann!ζn\hat\varphi(\zeta) = \sum_{n \ge 0} \frac{a_n}{n!} \zeta^n is Ω\Omega-continuable. The functions φ^\hat\varphi obtained in this way are the resurgent functions in the Borel plane. The series φ~\tilde\varphi is a resurgent series if it is Ω\Omega-resurgent for some closed discrete Ω\Omega.

Remarks

Example: φ~(z)=∑n≥0(−1)nn! z−n−1\tilde\varphi(z) = \sum_{n \ge 0} (-1)^n n! \, z^{-n-1} has Borel transform ∑n≥0(−1)nζn=11+ζ\sum_{n \ge 0} (-1)^n \zeta^n = \frac{1}{1 + \zeta}. This converges on the unit disc and extends to a holomorphic function on ℂ∖{−1}\C \setminus \set{-1}, so φ~\tilde\varphi is {−1}\set{-1}-resurgent. Non-example: ∑n≥0(n!)2z−n−1\sum_{n \ge 0} (n!)^2 z^{-n-1} is not resurgent, because its Borel transform ∑n≥0n! ζn\sum_{n \ge 0} n! \, \zeta^n has radius of convergence 00, so clause 1 fails. Every convergent series ∑anz−n−1\sum a_n z^{-n-1} is resurgent, since then φ^\hat\varphi is entire. Écalle's general notion allows the singular set to depend on the path of continuation; it is called endless continuability, and the Ω\Omega-continuable germs are a special case. The Borel transform of the formal product of two resurgent series is the convolution of their Borel transforms, and Écalle showed that resurgence is preserved under this product.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Analytic continuation, concept, no evidence levelAnalytic continuationconceptBorel summation, concept, no evidence levelBorel summationconceptComplex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionResurgent function, definition, no evidence levelResurgent functiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Analytic continuation concept
  2. Borel summation concept
  3. Complex numbers object
  4. Real numbers object
  5. Set bounded above definition
  6. Real and imaginary parts definition
  7. Supremum definition
  8. Completeness axiom statement
  9. Existence of square roots of nonnegative reals statement
  10. Square root definition
  11. Modulus of a complex number definition
  12. Complex differentiability definition
  13. Holomorphic function definition
  14. Resurgent function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Resurgent_function
About this page
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Kinddefinition
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MSC34M30, 30E15, 40G10
DefinesResurgent function
UsesHolomorphic function, Analytic continuation, Borel summation
Changed2 November 2026

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