Gamma

Gamma[z]

is the Euler gamma function .

Gamma[a,z]

is the incomplete gamma function .

Gamma[a,z₀,z₁]

is the generalized incomplete gamma function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The gamma function satisfies .
The incomplete gamma function satisfies .
The generalized incomplete gamma function is given by the integral .
Note that the arguments in the incomplete form of Gamma are arranged differently from those in the incomplete form of Beta.
Gamma[z] has no branch cut discontinuities.
Gamma[a,z] has a branch cut discontinuity in the complex z plane running from to .
For certain special arguments, Gamma automatically evaluates to exact values.
Gamma can be evaluated to arbitrary numerical precision.
Gamma automatically threads over lists.

Examples

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Basic Examples (8)

Integer values:

Half-integer values:

Evaluate numerically for complex arguments:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope (49)

Numerical Evaluation (4)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate Gamma efficiently at high precision:

Gamma threads elementwise over lists:

Specific Values (5)

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:

Visualization (3)

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:

Function Properties (10)

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 :

Differentiation (4)

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 :

Integration (3)

Indefinite integral of the incomplete gamma function:

Indefinite integrals of a product involving the incomplete gamma function:

Numerical approximation of a definite integral $int_1^2TemplateBox[{x}, Gamma]dx$ :

Series Expansions (6)

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:

Integral Transforms (4)

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:

Function Identities and Simplifications (5)

For positive integers :

Use FullSimplify to simplify gamma functions:

The Euler gamma function basic relation, :

The Euler gamma function of a double argument, $TemplateBox[{{2, , x}}, Gamma]=(2^(2 x-1))/(sqrt(pi)) TemplateBox[{x}, Gamma] TemplateBox[{{x, +, {1, /, 2}}}, Gamma]$ :

Relation to the incomplete gamma function:

Function Representations (5)

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:

Generalizations & Extensions (6)

Euler Gamma Function (3)

Gamma threads element-wise over lists:

Series expansion at poles:

Expansion at symbolically specified negative integers:

TraditionalForm formatting:

Incomplete Gamma Function (1)

Evaluate symbolically at integer and half‐integer orders:

Generalized Incomplete Gamma Function (2)

Evaluate symbolically at integer and half‐integer orders:

Series expansion at a generic point:

Applications (5)

Plot of the absolute value of Gamma in the complex plane:

Find the asymptotic expansion of ratios of gamma functions:

Volume of an ‐dimensional unit hypersphere:

Low‐dimensional cases:

Plot the volume of the unit hypersphere as a function of dimension:

Plot the real part of the incomplete gamma function over the parameter plane:

CDF of the ‐distribution:

Calculate the PDF:

Plot the CDF for different numbers of degrees of freedom:

Properties & Relations (7)

Use FullSimplify to simplify gamma functions:

Numerically find a root of a transcendental equation:

Sum expressions involving Gamma:

Generate from integrals, products, and limits:

Obtain Gamma as the solution of a differential equation:

Integrals:

Gamma can be represented as a DifferenceRoot:

Possible Issues (2)

Large arguments can give results too large to be computed explicitly:

Machine‐number inputs can give high‐precision results:

Neat Examples (2)

Nest Gamma over the complex plane:

Fractal from iterating Gamma:

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Gamma

Details

Examples

Basic Examples (8)