Math 2.0

Functions of a complex variable Definition

Wiman–Valiron theory

Builds on Entire functionTaylor series

Definition

Let f(z)=∑n≥0anznf(z) = \sum_{n \ge 0} a_n z^n be a transcendental entire function, that is, an entire function that is not a polynomial. Wiman–Valiron theory studies ff through the following functions of r>0r > 0:

  1. the maximum term μ(r)=max⁡n≥0∣an∣rn\mu(r) = \max_{n \ge 0} \abs{a_n} r^n;
  2. the central index ν(r)\nu(r), the largest integer nn with ∣an∣rn=μ(r)\abs{a_n} r^n = \mu(r).

Remarks

The maximum exists because ∣an∣rn→0\abs{a_n} r^n \to 0 as n→∞n \to \infty for every rr, by convergence of the Taylor series. The function ν(r)\nu(r) is nondecreasing, and ν(r)→∞\nu(r) \to \infty because ff is not a polynomial. Example: for f=ezf = e^z, an=1/n!a_n = 1/n! and the ratio of consecutive terms is r/(n+1)r/(n+1), so ν(r)=⌊r⌋\nu(r) = \lfloor r \rfloor and μ(r)=rν(r)/ν(r)!\mu(r) = r^{\nu(r)}/\nu(r)!. Non-example: for a polynomial of degree dd, ν(r)=d\nu(r) = d for all large rr, so it stays bounded; this is why the theory requires ff transcendental. The Cauchy estimates give μ(r)≤M(r)\mu(r) \le M(r), where M(r)=max⁡∣z∣=r∣f(z)∣M(r) = \max_{\abs{z} = r} \abs{f(z)}. The main result states that, for rr outside a set EE with ∫Edr/r<∞\int_E dr/r < \infty and zrz_r with ∣zr∣=r\abs{z_r} = r and ∣f(zr)∣=M(r)\abs{f(z_r)} = M(r), one has f(k)(zr)=(ν(r)/zr)kf(zr) (1+o(1))f^{(k)}(z_r) = \left(\nu(r)/z_r\right)^k f(z_r)\,(1 + o(1)) as r→∞r \to \infty, for each fixed kk.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectLimit point of a set, definition, no evidence levelLimit point of a setdefinitionReal numbers, object, no evidence levelReal numbersobjectSequence, definition, no evidence levelSequencedefinitionSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSupremum, definition, no evidence levelSupremumdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionSeries, definition, no evidence levelSeriesdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionPower series, definition, no evidence levelPower seriesdefinitionExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementDerivative function, definition, no evidence levelDerivative functiondefinitionSquare root, definition, no evidence levelSquare rootdefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionSmooth function, definition, no evidence levelSmooth functiondefinitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionTaylor series, definition, no evidence levelTaylor seriesdefinitionEntire function, definition, no evidence levelEntire functiondefinitionWiman–Valiron theory, definition, no evidence levelWiman–Valiron theorydefinition
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. Limit point of a set definition
  3. Real numbers object
  4. Sequence definition
  5. Set bounded above definition
  6. Real and imaginary parts definition
  7. Limit of a function definition
  8. Real sequence definition
  9. Supremum definition
  10. Continuity at a point definition
  11. Derivative definition
  12. Series definition
  13. Completeness axiom statement
  14. Continuous function definition
  15. Differentiable function definition
  16. Power series definition
  17. Existence of square roots of nonnegative reals statement
  18. Derivative function definition
  19. Square root definition
  20. Higher derivative definition
  21. Modulus of a complex number definition
  22. Taylor polynomial definition
  23. Continuously differentiable function definition
  24. Complex differentiability definition
  25. Smooth function definition
  26. Holomorphic function definition
  27. Taylor series definition
  28. Entire function definition
  29. Wiman–Valiron theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wiman%E2%80%93Valiron_theory
About this page
TagK7TYSAWGPC
URIm2://K7TYSAWGPC
Kinddefinition
WritingW0 stub
MSC30D20, 30D15
DefinesWiman–Valiron theory
UsesEntire function, Taylor series
Changed2 November 2026

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