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.
Gamma can be used with Interval and CenteredInterval objects. »

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 (50)

Numerical Evaluation (5)

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:

Gamma can be used with Interval and CenteredInterval objects:

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 (9)

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:

Compute derivatives of the Gamma function with the BellY polynomial:

Compute as a limit of Gamma functions at Infinity:

Expectation value of the square root of a quadratic form over a normal distribution:

Compare to closed-form result in terms of Gamma and CarlsonRG:

Represent Zeta in terms of Integrate and the Gamma function:

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)