Math 2.0

Functions of a complex variable Theorem

Borel–Ritt theorem

Builds on Holomorphic functionFormal power seriesAsymptotic expansion

Statement

Let 0<r≤∞0 < r \le \infty and α<β\alpha < \beta be real numbers, and let S={z∈ℂ:0<∣z∣<r, α<arg⁡z<β}S = \set{z \in \C : 0 < \abs{z} < r, \ \alpha < \arg z < \beta} be an open sector with vertex 00, of any finite opening β−α\beta - \alpha. Let ∑n≥0anzn\sum_{n \ge 0} a_n z^n be a formal power series with complex coefficients. Then there is a holomorphic function ff on SS with the following property. On every closed subsector {z:0<∣z∣≤ρ, α′≤arg⁡z≤β′}\set{z : 0 < \abs{z} \le \rho, \ \alpha' \le \arg z \le \beta'} with ρ<r\rho < r and α<α′<β′<β\alpha < \alpha' < \beta' < \beta, f(z)∼∑n≥0anznas z→0,f(z) \sim \sum_{n \ge 0} a_n z^n \quad \text{as } z \to 0, in the sense of asymptotic expansions in powers of zz.

Remarks

Idea of the proof: after a rotation, one picks γ>0\gamma > 0 so small that Re⁡(z−γ)>0\operatorname{Re}(z^{-\gamma}) > 0 on SS for a branch of z−γz^{-\gamma}. Then one sets f(z)=∑nanzn(1−e−bnz−γ)f(z) = \sum_{n} a_n z^n \left(1 - e^{-b_n z^{-\gamma}}\right) with positive numbers bnb_n that tend to 00 fast. Each factor 1−e−bnz−γ1 - e^{-b_n z^{-\gamma}} is bounded on SS and at most bn∣z∣−γb_n \abs{z}^{-\gamma} in absolute value, which makes the series converge and keeps the tail small near 00. The function ff is never unique: adding e−z−γe^{-z^{-\gamma}} for a suitable γ\gamma does not change the expansion. The real analogue, with smooth functions on an interval, is Borel's lemma. If the opening of SS is larger than 2π2\pi, SS is understood on the Riemann surface of the logarithm, and the theorem still holds.

Going deeper

For a student

The theorem shows that the map from functions with an asymptotic expansion on a sector to formal power series is onto. Its kernel consists of functions that are flat at 00 in the sector, such as e−1/ze^{-1/z} on ∣arg⁡z∣<π/2\abs{\arg z} < \pi/2. To recover uniqueness one needs stronger conditions, such as Gevrey estimates on a sector of opening larger than π/k\pi/k. This is the starting point of kk-summability and of Borel summation for divergent solutions of differential equations.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Big O notation, definition, no evidence levelBig O notationdefinitionComplex numbers, object, no evidence levelComplex numbersobjectFormal power series, definition, no evidence levelFormal power seriesdefinitionReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionLittle o notation, definition, no evidence levelLittle o notationdefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionAsymptotic expansion, definition, no evidence levelAsymptotic expansiondefinitionCompleteness 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 functiondefinitionBorel–Ritt theorem, statement, no evidence levelBorel–Ritt theoremstatement
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. Big O notation definition
  2. Complex numbers object
  3. Formal power series definition
  4. Real numbers object
  5. Set bounded above definition
  6. Little o notation definition
  7. Real and imaginary parts definition
  8. Supremum definition
  9. Asymptotic expansion definition
  10. Completeness axiom statement
  11. Existence of square roots of nonnegative reals statement
  12. Square root definition
  13. Modulus of a complex number definition
  14. Complex differentiability definition
  15. Holomorphic function definition
  16. Borel–Ritt theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Borel%27s_lemma
About this page
TagC2FFQAPJ9F
URIm2://C2FFQAPJ9F
Kindstatement, theorem
WritingW0 stub
MSC30E15, 41A60, 34M30
StatusThis copy of the site has no build data. Run m2 status C2FFQAPJ9F.
AboutAsymptotic expansion
UsesHolomorphic function, Formal power series, Asymptotic expansion
Changed2 November 2026

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