Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate Gamma efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Gamma function using MatrixFunction:

Singular points of Gamma:

Values at infinity:

Find a local minimum as a root of :

Evaluate the incomplete gamma function symbolically at integer and half‐integer orders:

Evaluate the generalized incomplete gamma function symbolically at half‐integer orders:

Plot the Euler gamma function:

Plot the real part of :

Plot the imaginary part of :

Plot the incomplete gamma function for integer and half-integer orders:

Real domain of the complete Euler gamma function:

Complex domain:

Domain of the incomplete gamma functions:

The gamma function achieves all nonzero values on the reals:

The incomplete gamma function achieves all positive real values for real inputs:

On the complexes, however, it achieves all nonzero values:

The incomplete gamma function has the restricted range :

The Euler gamma function has the mirror property :

The complete gamma function is a meromorphic, nonanalytic function:

is analytic in for positive integer :

But in general, it is neither an analytic nor a meromorphic function:

has both singularities and discontinuities on the non-positive integers:

is neither non-increasing nor non-decreasing:

is a non-increasing function of when is a positive, odd integer:

But in general, it is neither non-increasing nor non-decreasing:

is not injective:

is an injective function of for noninteger :

For integer , it may or may not be injective in :

is not surjective:

is also not surjective:

Visualize for :

is neither non-negative nor non-positive:

is non-negative for positive odd :

In general, it is neither non-negative nor non-positive:

is neither convex nor concave:

is convex on its real domain for :

It is in general neither convex nor concave for other values of :

First derivative of the Euler gamma function:

First derivative of the incomplete gamma function:

Higher derivatives of the Euler gamma function:

Higher derivatives of the incomplete gamma function for an order :

Indefinite integral of the incomplete gamma function:

Indefinite integrals of a product involving the incomplete gamma function:

Numerical approximation of a definite integral :

Taylor expansion for the Euler gamma function around :

Plot the first three approximations for the Euler gamma function around :

Series expansion at infinity for the Euler gamma function (Stirling approximation):

Give the result for an arbitrary symbolic direction:

Series expansion for the incomplete gamma function at a generic point:

Series expansion for the incomplete gamma function at infinity:

Series expansion for the generalized incomplete gamma function at a generic point:

Gamma can be applied to a power series:

Compute the Laplace transform of the incomplete gamma function using LaplaceTransform:

InverseLaplaceTransform of the incomplete gamma function:

MellinTransform of the incomplete gamma function:

InverseMellinTransform of the Euler gamma function:

For positive integers :

Use FullSimplify to simplify gamma functions:

The Euler gamma function basic relation, :

The Euler gamma function of a double argument, :

Relation to the incomplete gamma function:

Integral representation of the Euler gamma function:

Integral representation of the incomplete gamma function:

The incomplete gamma function can be represented in terms of MeijerG:

The incomplete gamma function can be represented as a DifferentialRoot:

TraditionalForm formatting:

Gamma threads element-wise over lists:

Series expansion at poles:

Expansion at symbolically specified negative integers:

TraditionalForm formatting:

Evaluate symbolically at integer and half‐integer orders:

Evaluate symbolically at integer and half‐integer orders:

Series expansion at a generic point: