Math 2.0

Special functions Definition

Reciprocal gamma function

Builds on Gamma functionEntire function

Definition

Let Γ\Gamma be the gamma function, extended to a meromorphic function on ℂ\C with poles at 0,−1,−2,…0, -1, -2, \ldots and no zeros. The reciprocal gamma function is the function ℂ→ℂ\C \to \C given by 1Γ(z)={1/Γ(z)if z∉{0,−1,−2,…},0if z∈{0,−1,−2,…}.\frac{1}{\Gamma}(z) = \begin{cases} 1/\Gamma(z) & \text{if } z \notin \set{0, -1, -2, \ldots}, \\ 0 & \text{if } z \in \set{0, -1, -2, \ldots}. \end{cases}

Remarks

It is an entire function: near a pole −n-n of Γ\Gamma, the quotient 1/Γ1/\Gamma has a removable singularity with value 00, because the pole is simple. Its zeros are exactly 0,−1,−2,…0, -1, -2, \ldots, and they are simple, with derivative (−1)nn!(-1)^n n! at −n-n, the reciprocal of the residue of Γ\Gamma. Example: at z=3z = 3 the value is 1/Γ(3)=1/2!=121/\Gamma(3) = 1/2! = \tfrac12. Non-example: Γ\Gamma itself is not entire, since it is not defined at 00 and ∣Γ(z)∣→∞\abs{\Gamma(z)} \to \infty as z→0z \to 0. It is an entire function of order 11 and of maximal type.

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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 functiondefinitionEntire function, definition, no evidence levelEntire functiondefinitionReciprocal gamma function, definition, no evidence levelReciprocal gamma func…definition
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.
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  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. Entire function definition
  34. Reciprocal gamma function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Reciprocal_gamma_function
About this page
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URIm2://YGK9CD7SME
Kinddefinition
WritingW0 stub
MSC33B15, 30D15
DefinesReciprocal gamma function
UsesGamma function, Entire function

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