Math 2.0

Special functions Gamma function and related functions Theorem

Gamma function has no zeros

Builds on Gamma functionMeromorphic function

Statement

Let Γ\Gamma be the gamma function, extended to a meromorphic function on ℂ\C. For every z∈ℂ∖{0,−1,−2,…}z \in \C \setminus \set{0, -1, -2, \ldots}, Γ(z)≠0\Gamma(z) \ne 0.

Remarks

For z∉ℤz \notin \Z, the reflection formula gives Γ(z)Γ(1−z)=π/sin⁡(πz)\Gamma(z)\Gamma(1 - z) = \pi/\sin(\pi z), and the right side is finite and nonzero, so Γ(z)≠0\Gamma(z) \ne 0. For a positive integer nn, Γ(n)=(n−1)!≠0\Gamma(n) = (n-1)! \ne 0. The points 0,−1,−2,…0, -1, -2, \ldots are excluded because Γ\Gamma has poles there, not values. A consequence is that the reciprocal gamma function 1/Γ1/\Gamma is entire, with zeros exactly at 0,−1,−2,…0, -1, -2, \ldots. In Mathlib the statement is Complex.Gamma_ne_zero.

Going deeper

For a student

Because Γ\Gamma has no zeros, its logarithmic derivative, the digamma function ψ=Γ′/Γ\psi = \Gamma'/\Gamma, is meromorphic with poles only at 0,−1,−2,…0, -1, -2, \ldots. It also allows a holomorphic logarithm of Γ\Gamma on the slit plane, the log-gamma function.

Reading path

To understand this page from the ground up, read these 36 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 setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionReal 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 functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSupremum, definition, no evidence levelSupremumdefinitionLimit 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 sequencedefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementRiemann integral, definition, no evidence levelRiemann integraldefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…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…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementComplex differentiability, definition, no evidence levelComplex differentiabi…definitionComparison test for improper integrals, statement, no evidence levelComparison test for i…statementHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionGamma function, definition, no evidence levelGamma functiondefinitionIsolated singularity, definition, no evidence levelIsolated singularitydefinitionPole, definition, no evidence levelPoledefinitionMeromorphic function, definition, no evidence levelMeromorphic functiondefinitionGamma function has no zeros, statement, no evidence levelGamma function has no…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. Limit point of a set definition
  3. Partition of an interval definition
  4. Real numbers object
  5. Sequence definition
  6. Set bounded above definition
  7. Real and imaginary parts definition
  8. Limit of a function definition
  9. Mesh of a partition definition
  10. Riemann sum definition
  11. Real sequence definition
  12. Supremum definition
  13. Limit at infinity definition
  14. One-sided limit definition
  15. Riemann integrable function definition
  16. Bounded sequence definition
  17. Limit of a sequence definition
  18. Monotone sequence definition
  19. Completeness axiom statement
  20. Riemann integral definition
  21. Monotone convergence theorem for sequences statement
  22. Existence of square roots of nonnegative reals statement
  23. Definite integral definition
  24. Improper integral definition
  25. One-sided limits of a monotone function exist statement
  26. Square root definition
  27. Modulus of a complex number definition
  28. Monotonicity of the integral statement
  29. Complex differentiability definition
  30. Comparison test for improper integrals statement
  31. Holomorphic function definition
  32. Gamma function definition
  33. Isolated singularity definition
  34. Pole definition
  35. Meromorphic function definition
  36. Gamma function has no zeros statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gamma_function
About this page
TagWBS4NM1TFC
URIm2://WBS4NM1TFC
Kindstatement, theorem
WritingW0 stub
MSC33B15, 30D30
StatusThis copy of the site has no build data. Run m2 status WBS4NM1TFC.
AboutGamma function
UsesGamma function, Meromorphic function

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