Math 2.0

Functions of a complex variable Complex integration Corollary

Cauchy's estimates

Builds on Holomorphic functionCauchy's integral formula for derivativesEstimation lemma

Statement

Let U⊆ℂU \subseteq \mathbb{C} be an open set and let f:U→ℂf : U \to \mathbb{C} be holomorphic on UU. Let c∈ℂc \in \mathbb{C} and r>0r > 0 be such that the closed disc {z:∣z−c∣≤r}\{z : |z - c| \le r\} is contained in UU. Let M≥0M \ge 0 be a real number with ∣f(z)∣≤M|f(z)| \le M for each zz with ∣z−c∣=r|z - c| = r. Then for each integer n≥0n \ge 0, ∣f(n)(c)∣≤n! Mrn.|f^{(n)}(c)| \le \frac{n! \, M}{r^n}.

Remarks

Proof: apply the estimation lemma to the formula for the derivatives with a=ca = c. The circle has length 2πr2 \pi r, and ∣f(z)/(z−c)n+1∣≤M/rn+1|f(z) / (z - c)^{n+1}| \le M / r^{n+1} on it. For n=1n = 1, the estimate is ∣f′(c)∣≤M/r|f'(c)| \le M / r. If f(z)=∑an(z−c)nf(z) = \sum a_n (z - c)^n is the Taylor series, the estimate is ∣an∣≤M/rn|a_n| \le M / r^n.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementEstimation lemma, statement, no evidence levelEstimation lemmastatementExistence 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 functiondefinitionCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statementHolomorphic functions are analytic, statement, no evidence levelHolomorphic functions…statementHolomorphic functions are infinitely differentiable, statement, no evidence levelHolomorphic functions…statementCauchy's integral formula for derivatives, statement, no evidence levelCauchy's integral for…statementCauchy's estimates, statement, no evidence levelCauchy's estimatesstatement
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. Set bounded above definition
  4. Complex exponential definition
  5. Contour integral definition
  6. Power series in the complex plane definition
  7. Real and imaginary parts definition
  8. Supremum definition
  9. Circle integral definition
  10. Disc of convergence definition
  11. Completeness axiom statement
  12. Estimation lemma statement
  13. Existence of square roots of nonnegative reals statement
  14. Square root definition
  15. Modulus of a complex number definition
  16. Complex differentiability definition
  17. Holomorphic function definition
  18. Cauchy's integral formula statement
  19. Power series are holomorphic in the disc of convergence statement
  20. Holomorphic functions are analytic statement
  21. Holomorphic functions are infinitely differentiable statement
  22. Cauchy's integral formula for derivatives statement
  23. Cauchy's estimates statement

Where this leads

12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula, Consequences
About this page
Tag0JJRENWMP6
URIm2://0JJRENWMP6
Aliasescauchy-inequalities, cauchy-estimate
Kindstatement, corollary
WritingW0 stub
MSC30A10
StatusThis copy of the site has no build data. Run m2 status 0JJRENWMP6.
UsesHolomorphic function, Cauchy's integral formula for derivatives, Estimation lemma

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