• ← 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 gamma function on the complex number z.

Gamma[z, x]

is the upper incomplete gamma function.

Gamma[z, $x_0$, $x_1$]

is equivalent to Gamma[z, $x_0$] - Gamma[z, $x_1$].

Gamma[z] is equivalent to (z - 1)!:

• • Simplify[Gamma[z] - (z - 1)!]

Exact arguments:

• • Gamma[8]

• • Gamma[1/2]

• • Gamma[1, x]

• • Gamma[0, x]

Numeric arguments:

• • Gamma[123.78]

• • Gamma[1. + I]

Both Gamma and Factorial functions are continuous:

• • Plot[{Gamma[x], x!}, {x, 0, 4}]