Math 2.0

Special functions Definition

Gamma function

Builds on Improper integralComparison test for improper integrals

Definition

For a real number s>0s > 0, the gamma function is Γ(s)=∫0∞ts−1e−t dt.\Gamma(s) = \int_0^\infty t^{s-1} e^{-t} \, dt.

Remarks

The improper integral converges at 00 because s>0s > 0, and at ∞\infty because e−te^{-t} decays fast. Both claims follow from the comparison test. Integration by parts gives Γ(s+1)=s Γ(s)\Gamma(s + 1) = s \, \Gamma(s). Also Γ(1)=1\Gamma(1) = 1, so Γ\Gamma extends the factorial. The function extends to all real numbers except 0,−1,−2,…0, -1, -2, \dots, by the recursion. Mathlib defines this extension as Real.Gamma. The notation Γ\Gamma is due to Adrien-Marie Legendre.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

Show the path as a graph
Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, 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 functiondefinition
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. Sequence definition
  4. Limit of a function definition
  5. Mesh of a partition definition
  6. Riemann sum definition
  7. Real sequence definition
  8. Limit at infinity definition
  9. One-sided limit definition
  10. Riemann integrable function definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Riemann integral definition
  15. Monotone convergence theorem for sequences statement
  16. Definite integral definition
  17. Improper integral definition
  18. One-sided limits of a monotone function exist statement
  19. Monotonicity of the integral statement
  20. Comparison test for improper integrals statement
  21. Gamma function definition

Where this leads

29 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gamma_function, Main definition
About this page
Tag0V3FZMVB4K
URIm2://0V3FZMVB4K
Kinddefinition
WritingW1 stated
MSC33B15
DefinesGamma function
UsesImproper integral, Comparison test for improper integrals

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by

Results about this