# 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(44)

### 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:

### 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 halfinteger orders:

Evaluate the generalized incomplete gamma function symbolically at halfinteger 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(5)

Real domain of the Euler gamma function:

Complex domain:

The incomplete gamma function for an integer order takes all real positive values:

The range for complex values:

The incomplete gamma function for takes all real values from the interval :

The incomplete gamma function for takes all real positive values:

The Euler gamma function has the mirror property :

### 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 :

### 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, :

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:

## Generalizations & Extensions(6)

### Euler Gamma Function(3)

Series expansion at poles:

Expansion at symbolically specified negative integers:

### Incomplete Gamma Function(1)

Evaluate symbolically at integer and halfinteger orders:

### Generalized Incomplete Gamma Function(2)

Evaluate symbolically at integer and halfinteger 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:

Lowdimensional 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: Machinenumber inputs can give highprecision results:

## Neat Examples(2)

Nest Gamma over the complex plane:

Fractal from iterating Gamma:

Introduced in 1988
(1.0)