Math 2.0

Functions of a complex variable Theorem

Power series are holomorphic in the disc of convergence

Builds on Power series in the complex planeDisc of convergenceHolomorphic function

Statement

Let ∑n≥0an(z−c)n\sum_{n \ge 0} a_n (z - c)^n be a power series with radius of convergence R>0R > 0, and let D={z:∣z−c∣<R}D = \{z : |z - c| < R\} be its disc of convergence. Let f(z)=∑n=0∞an(z−c)nf(z) = \sum_{n=0}^{\infty} a_n (z - c)^n for z∈Dz \in D. Then ff is holomorphic on DD, and for each z∈Dz \in D, f′(z)=∑n=1∞nan(z−c)n−1.f'(z) = \sum_{n=1}^{\infty} n a_n (z - c)^{n-1}. The series for f′f' has the same radius of convergence RR.

Remarks

By induction, ff has complex derivatives of all orders on DD, and an=f(n)(c)/n!a_n = f^{(n)}(c) / n!. So the power series is the Taylor series of ff at cc. For example, the complex exponential has R=∞R = \infty, so it is entire. The converse is also true: holomorphic functions are analytic.

Reading path

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionPower 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 levelSupremumdefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness 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 functiondefinitionPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statement
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. Power series in the complex plane definition
  5. Real and imaginary parts definition
  6. Supremum definition
  7. Disc of convergence 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. Power series are holomorphic in the disc of convergence statement

Where this leads

75 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Power_series, Differentiation and integration
  2. Web https://en.wikipedia.org/wiki/Power_series, Analytic functions
About this page
Tag0YMP521ZR5
URIm2://0YMP521ZR5
Aliasestermwise-differentiation-of-power-series
Kindstatement, theorem
WritingW0 stub
MSC30B10
StatusThis copy of the site has no build data. Run m2 status 0YMP521ZR5.
UsesPower series in the complex plane, Disc of convergence, Holomorphic function

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