• ← Factorial2
• → LogGamma

Gamma

Gamma function (SymPy, mpmath, WMA)

The gamma function is one commonly used extension of the factorial function applied to complex numbers, and is defined for all complex numbers except the non-positive integers.

Gamma[z]

is the Euler gamma function, $\Gamma(z)$ on the complex number z.

Gamma[a, z]

is the upper incomplete gamma function, $\Gamma(a, z)$.

Gamma[a, $z_0$, $z_1$]

is the generalized incomplete gamma function $\Gamma[a, z_0] - \Gamma(a, z_1)$.

Gamma[z] is equivalent to $(z - 1)!$:

• • Simplify[Gamma[z] - (z - 1)!]

Examples of using Gamma with exact numeric arguments:

• • Gamma[8]

• • Gamma[1/2]

• • Gamma[123.78]

• • Gamma[1. + I]

Examples of Gamma with symbolic arguments:

• • Gamma[1, x]

• • Gamma[0, x]

Both Gamma and Factorial functions are continuous:

• • Plot[{Gamma[x], x!}, {x, 0, 4}]