Math 2.0

Several complex variables and analytic spaces Holomorphic functions of several variables Lemma

Osgood's lemma

Builds on Holomorphic functionCauchy's integral formulaPower series

Statement

Let n≥1n \ge 1, let U⊆ℂnU \subseteq \C^n be an open set, and let f ⁣:U→ℂf \colon U \to \C be continuous. Assume that ff is holomorphic in each variable separately: for each jj and each fixed value of the other variables, zj↦f(z1,…,zn)z_j \mapsto f(z_1, \dots, z_n) is holomorphic. Then ff is holomorphic on UU. That is, near each point c∈Uc \in U, ff is the sum of a convergent power series in z1−c1,…,zn−cnz_1 - c_1, \dots, z_n - c_n.

Remarks

Proof idea: on a closed polydisc ∏j{∣zj−cj∣≤rj}\prod_j \set{\abs{z_j - c_j} \le r_j} in UU, apply the one-variable Cauchy integral formula in each variable in turn. Continuity makes the iterated integral an integral over the torus ∏j{∣ζj−cj∣=rj}\prod_j \set{\abs{\zeta_j - c_j} = r_j}. Expanding the kernel ∏j(ζj−zj)−1\prod_j (\zeta_j - z_j)^{-1} as a geometric series gives the power series. Hartogs's theorem on separate analyticity shows that the continuity hypothesis can be dropped, but its proof is much harder. For real variables the analogue fails: xy/(x2+y2)xy/(x^2 + y^2), set to 00 at the origin, is real-analytic in each variable but not continuous.

Going deeper

For a student

Osgood's lemma is often the first step in a course on several complex variables. With it, the Cauchy formula on polydiscs gives power series expansions, Cauchy estimates and Weierstrass convergence theorems in ℂn\C^n. It also lets one define holomorphic functions of several variables either by power series or by separate holomorphy plus continuity.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSequence, definition, no evidence levelSequencedefinitionSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionReal sequence, definition, no evidence levelReal sequencedefinitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementSeries, definition, no evidence levelSeriesdefinitionExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementPower series, definition, no evidence levelPower seriesdefinitionSquare 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 functiondefinitionCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementOsgood's lemma, statement, no evidence levelOsgood's lemmastatement
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. Complex numbers object
  2. Real numbers object
  3. Sequence definition
  4. Set bounded above definition
  5. Complex exponential definition
  6. Contour integral definition
  7. Real and imaginary parts definition
  8. Real sequence definition
  9. Supremum definition
  10. Circle integral definition
  11. Completeness axiom statement
  12. Series definition
  13. Existence of square roots of nonnegative reals statement
  14. Power series definition
  15. Square root definition
  16. Modulus of a complex number definition
  17. Complex differentiability definition
  18. Holomorphic function definition
  19. Cauchy's integral formula statement
  20. Osgood's lemma statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Osgood%27s_lemma
About this page
TagNBFNXA2WFM
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Kindstatement, lemma
WritingW0 stub
MSC32A10
StatusThis copy of the site has no build data. Run m2 status NBFNXA2WFM.
AboutHolomorphic function
UsesHolomorphic function, Cauchy's integral formula, Power series
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Changed2 November 2026

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