Math 2.0

Special functions Gamma function and related functions Theorem

Hölder's theorem

Builds on Gamma functionPolynomial ring

Statement

Let Γ\Gamma be the gamma function, as a holomorphic function on ℂ∖{0,−1,−2,…}\C \setminus \set{0, -1, -2, \ldots}. Let n≥0n \ge 0 be an integer and let P∈ℂ[Y,X0,X1,…,Xn]P \in \C[Y, X_0, X_1, \ldots, X_n] be a polynomial in n+2n + 2 variables (an element of a polynomial ring). If P(z,Γ(z),Γ′(z),…,Γ(n)(z))=0P\big(z, \Gamma(z), \Gamma'(z), \ldots, \Gamma^{(n)}(z)\big) = 0 for all zz in the domain of Γ\Gamma, then P=0P = 0.

Remarks

In words, Γ\Gamma satisfies no algebraic differential equation whose coefficients are polynomials in zz; one says that Γ\Gamma is differentially transcendental. One proof assumes a nonzero PP of least possible degree and uses the functional equation Γ(z+1)=zΓ(z)\Gamma(z + 1) = z\Gamma(z) to build from it a nonzero polynomial relation of smaller degree, a contradiction. In particular Γ\Gamma is not a holonomic function, since a linear differential equation with polynomial coefficients is a special algebraic one. By contrast, the exponential and many other special functions, such as Bessel functions, do satisfy algebraic differential equations. The theorem is named after Otto Hölder; it is not the same as Hölder's theorem on Archimedean groups.

Going deeper

For a student

The theorem shows that the gamma function is not a solution of any of the classical differential equations of mathematical physics. This contrasts with the hypergeometric and Bessel functions, which are defined by linear differential equations, and explains why Γ\Gamma is usually studied through functional equations, integrals and products instead.

Reading path

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

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionPolynomial ring, concept, no evidence levelPolynomial ringconceptSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit at infinity, definition, no evidence levelLimit at infinitydefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementDefinite integral, definition, no evidence levelDefinite integraldefinitionImproper integral, definition, no evidence levelImproper integraldefinitionOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementComparison test for improper integrals, statement, no evidence levelComparison test for i…statementGamma function, definition, no evidence levelGamma functiondefinitionHölder's theorem, statement, no evidence levelHölder's 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. Limit point of a set definition
  2. Partition of an interval definition
  3. Polynomial ring concept
  4. Sequence definition
  5. Limit of a function definition
  6. Mesh of a partition definition
  7. Riemann sum definition
  8. Real sequence definition
  9. Limit at infinity definition
  10. One-sided limit definition
  11. Riemann integrable function definition
  12. Bounded sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Riemann integral definition
  16. Monotone convergence theorem for sequences statement
  17. Definite integral definition
  18. Improper integral definition
  19. One-sided limits of a monotone function exist statement
  20. Monotonicity of the integral statement
  21. Comparison test for improper integrals statement
  22. Gamma function definition
  23. Hölder's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/H%C3%B6lder%27s_theorem
About this page
TagSF3X43X837
URIm2://SF3X43X837
Kindstatement, theorem
WritingW0 stub
MSC33B15, 12H05
StatusThis copy of the site has no build data. Run m2 status SF3X43X837.
AboutGamma function
UsesGamma function, Polynomial ring
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