# 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 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 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(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 :

### 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(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:

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:

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 with the 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:

Machinenumber inputs can give highprecision results:

## Neat Examples(3)

Nest Gamma over the complex plane:

Fractal from iterating Gamma:

Plot the Riemann surface of :

Wolfram Research (1988), Gamma, Wolfram Language function, https://reference.wolfram.com/language/ref/Gamma.html (updated 2022).

#### Text

Wolfram Research (1988), Gamma, Wolfram Language function, https://reference.wolfram.com/language/ref/Gamma.html (updated 2022).

#### CMS

Wolfram Language. 1988. "Gamma." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Gamma.html.

#### APA

Wolfram Language. (1988). Gamma. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Gamma.html

#### BibTeX

@misc{reference.wolfram_2024_gamma, author="Wolfram Research", title="{Gamma}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Gamma.html}", note=[Accessed: 20-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_gamma, organization={Wolfram Research}, title={Gamma}, year={2022}, url={https://reference.wolfram.com/language/ref/Gamma.html}, note=[Accessed: 20-June-2024 ]}