Math 2.0

statement theorem

The gamma function extends the factorial

Statement

For each s>0s > 0, the gamma function satisfies Γ(s+1)=s Γ(s)\Gamma(s + 1) = s \, \Gamma(s). For each natural number nn, Γ(n+1)=n!\Gamma(n + 1) = n!.

Remarks

Integration by parts on [ε,b][\varepsilon, b], with u=tsu = t^s and v=−e−tv = -e^{-t}, gives the recursion in the limit. The integral of e−te^{-t} gives Γ(1)=1\Gamma(1) = 1. Induction on nn then gives Γ(n+1)=n!\Gamma(n + 1) = n!. Also Γ(1/2)=π\Gamma(1/2) = \sqrt{\pi}, by the substitution t=x2t = x^2 and the Gaussian integral. Mathlib states it as Real.Gamma_nat_eq_factorial.

History

Leonhard Euler announced the integral representation in a letter to Christian Goldbach, dated 8 January 1730.

Explanations

L1 human

The recursion Γ(s+1)=s Γ(s)\Gamma(s + 1) = s \, \Gamma(s) is the same as the rule (n+1)!=(n+1)⋅n!(n + 1)! = (n + 1) \cdot n!. So the gamma function agrees with the factorial at the integers, shifted by one.

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  1. Limit point of a set definition
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  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
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  11. Continuity at a point definition
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  14. One-sided limit definition
  15. Riemann integrable function definition
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  19. Monotone sequence definition
  20. Series definition
  21. Continuous function definition
  22. Differentiable function definition
  23. Fermat's theorem on stationary points statement
  24. Riemann integral definition
  25. Sum of a finite geometric series statement
  26. Every sequence has a monotone subsequence statement
  27. Monotone convergence theorem for sequences statement
  28. Convergent sequence definition
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  33. Differentiable functions are continuous statement
  34. Derivative function definition
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  36. Definite integral definition
  37. One-sided limits of a monotone function exist statement
  38. Bolzano–Weierstrass theorem statement
  39. The sequence r^n tends to zero for |r| < 1 example
  40. Radius of convergence definition
  41. Convergent series definition
  42. Product rule statement
  43. Higher derivative definition
  44. Boundedness theorem statement
  45. Monotonicity of the integral statement
  46. A series of nonnegative terms converges iff its partial sums are bounded statement
  47. Absolute convergence definition
  48. Sum of a series definition
  49. Continuously differentiable function definition
  50. Extreme value theorem statement
  51. Comparison test for improper integrals statement
  52. Comparison test statement
  53. Geometric series example
  54. Cauchy product of absolutely convergent series statement
  55. Rolle's theorem statement
  56. Gamma function definition
  57. Ratio test statement
  58. Mean value theorem statement
  59. Ratio formula for the radius of convergence statement
  60. Second fundamental theorem of calculus statement
  61. The exponential series has infinite radius of convergence statement
  62. Integration by parts statement
  63. Exponential function definition
  64. Exponential of a sum statement
  65. The exponential function is its own derivative statement
  66. Integral of the exponential function statement
  67. Integral of exp(-x) over the half line example
  68. The gamma function extends the factorial statement

Sources

  1. Web https://en.wikipedia.org/wiki/Gamma_function, General

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