# The Γ function: Gamma

### 5.8.13 The Γ function: Gamma

The Gamma function is defined by

Γ(x)=

∫

+∞
0

e−ttx−1dt, if x>0

If x is a positive integer, Γ is computed by applying the recurrence:

Γ(x+1)=x*Γ(x), Γ(1)=1

Hence:

Γ(n+1)=n!

and the Gamma function is used to generalize the factorial (see Section 5.6.1).

The Gamma command computes the Gamma function.

- Gamma takes one argument:
  a, a number.
- Gamma(a) returns the value Γ(a).

#### Examples.

- Input:
  Gamma(5)
  Output:

  24

- Input:
  Gamma(0.7)
  Output:

  1.29805533265

- Input:
  Gamma(-0.3)
  Output:

  −4.32685110883

  Indeed: Gamma(0.7)=-0.3*Gamma(-0.3)
- Input:
  Gamma(-1.3)
  Output:

  3.32834700679

  Indeed Gamma(0.7)=-0.3*Gamma(-0.3)=(-0.3)*(-1.3)*Gamma(-1.3)