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Gamma Function

Definition

Self-study material

In 2026-27 the gamma function is the self-study section of Lecture 12. It is worth knowing as an application of improper integrals, but it is not developed in class.

The gamma function Γ ⁣:(0,+∞)→R\Gamma\colon (0,+\infty)\to \mathbb{R} is defined by an improper integral:

Γ(x)=∫0+∞tx−1e−t dt.\Gamma(x)=\int_0^{+\infty} t^{x-1}\mathrm{e}^{-t}\,\mathrm{d}t.

Interpretation

The gamma function extends the factorial from integers to all positive real arguments: Γ(n)=(n−1)!\Gamma(n)=(n-1)! for integers n≥1n\ge1, and between the integers Γ\Gamma is given by the same integral. Because of the shift by one, Γ(n+1)=n!\Gamma(n+1)=n!.

The gamma function for x from 0 to 5, falling steeply near 0 and then rising, passing through the points (1, 1), (2, 1), (3, 2), (4, 6) and (5, 24), the factorials.

The red dots are the factorials: Γ(1)=0!=1\Gamma(1)=0!=1, Γ(2)=1!=1\Gamma(2)=1!=1, Γ(3)=2\Gamma(3)=2, Γ(4)=6\Gamma(4)=6 and Γ(5)=4!=24\Gamma(5)=4!=24. The curve passes through them and also takes values in between, such as Γ(12)=π≈1.77\Gamma(\tfrac{1}{2})=\sqrt{\pi}\approx1.77. It has its minimum between 11 and 22, about 0.8860.886 near x≈1.46x\approx1.46, and grows without bound as x↓0x\downarrow0.

The link with factorials comes from integration by parts. For x>0x>0 it gives Γ(x+1)=x Γ(x)\Gamma(x+1)=x\,\Gamma(x), the same recursion as n!=n⋅(n−1)!n!=n\cdot(n-1)!, and Γ(1)=∫0∞e−t dt=1\Gamma(1)=\int_0^\infty\mathrm{e}^{-t}\,\mathrm{d}t=1; induction then gives Γ(n)=(n−1)!\Gamma(n)=(n-1)!. The integral is improper at ∞\infty and, for x<1x<1, also at 00; the domain (0,+∞)(0,+\infty) is where it converges. Near 00 the integrand behaves like tx−1t^{x-1}, which by the pp-test with p=1−xp=1-x is integrable exactly when x>0x>0. Near ∞\infty the factor e−t\mathrm{e}^{-t} outweighs any power, and comparison with t−2t^{-2} gives convergence.

A common slip is the shift: Γ(5)\Gamma(5) is 4!=244!=24, not 5!=1205!=120. The gamma function returns later in probability, in the formula of the gamma distribution.

Key Results

  • The gamma function satisfies Γ(n)=(n−1)!\Gamma(n)=(n-1)! for all integers n≥1.n\ge 1\text{.}
  • It generalizes factorial growth to real arguments.
  • The domain of the gamma function can be extended further.

Examples

  • Γ(1)=1.\Gamma(1)=1\text{.}
  • Γ(2)=1!=1.\Gamma(2)=1!=1\text{.}
  • Γ(3)=2!=2.\Gamma(3)=2!=2\text{.}

Prerequisites

Sources

Aliases: Euler's gamma function Symbols: gamma, infinity, integral, pi, real numbers, square root

Last updated: 27 October 2026 11:52