Gamma
✖
Gamma
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
open allclose allBasic Examples (8)Summary of the most common use cases

https://wolfram.com/xid/0dekoi-bhmfdi


https://wolfram.com/xid/0dekoi-iwos4u

Evaluate numerically for complex arguments:

https://wolfram.com/xid/0dekoi-sdmdt

Plot over a subset of the reals:

https://wolfram.com/xid/0dekoi-gvraz

Plot over a subset of the complexes:

https://wolfram.com/xid/0dekoi-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0dekoi-bpm

Series expansion at Infinity:

https://wolfram.com/xid/0dekoi-laddhh

Series expansion at a singular point:

https://wolfram.com/xid/0dekoi-gzfdvj

Scope (50)Survey of the scope of standard use cases
Numerical Evaluation (5)

https://wolfram.com/xid/0dekoi-l274ju


https://wolfram.com/xid/0dekoi-bqjj6g

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

https://wolfram.com/xid/0dekoi-doygxf

Evaluate Gamma efficiently at high precision:

https://wolfram.com/xid/0dekoi-di5gcr


https://wolfram.com/xid/0dekoi-bq2c6r

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

https://wolfram.com/xid/0dekoi-m94s2u


https://wolfram.com/xid/0dekoi-gnclzo


https://wolfram.com/xid/0dekoi-lmyeh7

Compute average-case statistical intervals using Around:

https://wolfram.com/xid/0dekoi-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0dekoi-thgd2

Or compute the matrix Gamma function using MatrixFunction:

https://wolfram.com/xid/0dekoi-o5jpo

Specific Values (5)
Singular points of Gamma:

https://wolfram.com/xid/0dekoi-nww7l


https://wolfram.com/xid/0dekoi-cgjpl5

Find a local minimum as a root of :

https://wolfram.com/xid/0dekoi-f2hrld


https://wolfram.com/xid/0dekoi-ioemx8

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

https://wolfram.com/xid/0dekoi-b2perf


https://wolfram.com/xid/0dekoi-3hv9k

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

https://wolfram.com/xid/0dekoi-c0e21

Visualization (3)
Plot the Euler gamma function:

https://wolfram.com/xid/0dekoi-ecj8m7


https://wolfram.com/xid/0dekoi-49yzz


https://wolfram.com/xid/0dekoi-cebvhb

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

https://wolfram.com/xid/0dekoi-bzo8ib

Function Properties (10)
Real domain of the complete Euler gamma function:

https://wolfram.com/xid/0dekoi-cl7ele


https://wolfram.com/xid/0dekoi-de3irc

Domain of the incomplete gamma functions:

https://wolfram.com/xid/0dekoi-njiu4n


https://wolfram.com/xid/0dekoi-k85ibq


https://wolfram.com/xid/0dekoi-gk8klh

The gamma function achieves all nonzero values on the reals:

https://wolfram.com/xid/0dekoi-ry64u6

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

https://wolfram.com/xid/0dekoi-evf2yr

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

https://wolfram.com/xid/0dekoi-fphbrc

The incomplete gamma function has the restricted range
:

https://wolfram.com/xid/0dekoi-14x2sd

The Euler gamma function has the mirror property :

https://wolfram.com/xid/0dekoi-heoddu

The complete gamma function is a meromorphic, nonanalytic function:

https://wolfram.com/xid/0dekoi-1kybw5


https://wolfram.com/xid/0dekoi-0r1ii1

is analytic in
for positive integer
:

https://wolfram.com/xid/0dekoi-2uxtb5

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

https://wolfram.com/xid/0dekoi-h5x4l2


https://wolfram.com/xid/0dekoi-oweui9

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

https://wolfram.com/xid/0dekoi-mdtl3h


https://wolfram.com/xid/0dekoi-mn5jws

is neither non-increasing nor non-decreasing:

https://wolfram.com/xid/0dekoi-ijfl94

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

https://wolfram.com/xid/0dekoi-w16tr0

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

https://wolfram.com/xid/0dekoi-nlz7s


https://wolfram.com/xid/0dekoi-poz8g

is an injective function of
for noninteger
:

https://wolfram.com/xid/0dekoi-u2ahs5


https://wolfram.com/xid/0dekoi-qv9pac

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

https://wolfram.com/xid/0dekoi-2fwk5p


https://wolfram.com/xid/0dekoi-n6ag2b


https://wolfram.com/xid/0dekoi-ctca0g


https://wolfram.com/xid/0dekoi-87rz8n


https://wolfram.com/xid/0dekoi-v22ou6


https://wolfram.com/xid/0dekoi-gp4j1j


https://wolfram.com/xid/0dekoi-hdm869

is neither non-negative nor non-positive:

https://wolfram.com/xid/0dekoi-84dui

is non-negative for positive odd
:

https://wolfram.com/xid/0dekoi-rj4658

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

https://wolfram.com/xid/0dekoi-mphroq

is neither convex nor concave:

https://wolfram.com/xid/0dekoi-8kku21

is convex on its real domain for
:

https://wolfram.com/xid/0dekoi-05oag1

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

https://wolfram.com/xid/0dekoi-7bjvvg

Differentiation (4)
First derivative of the Euler gamma function:

https://wolfram.com/xid/0dekoi-mmas49

First derivative of the incomplete gamma function:

https://wolfram.com/xid/0dekoi-n1p57j

Higher derivatives of the Euler gamma function:

https://wolfram.com/xid/0dekoi-nfbe0l


https://wolfram.com/xid/0dekoi-fxwmfc

Higher derivatives of the incomplete gamma function for an order :

https://wolfram.com/xid/0dekoi-bpenlh


https://wolfram.com/xid/0dekoi-dip326

Integration (3)
Indefinite integral of the incomplete gamma function:

https://wolfram.com/xid/0dekoi-bponid

Indefinite integrals of a product involving the incomplete gamma function:

https://wolfram.com/xid/0dekoi-ft0ejz


https://wolfram.com/xid/0dekoi-elcont

Numerical approximation of a definite integral :

https://wolfram.com/xid/0dekoi-b9jw7l

Series Expansions (6)
Taylor expansion for the Euler gamma function around :

https://wolfram.com/xid/0dekoi-ewr1h8

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

https://wolfram.com/xid/0dekoi-binhar

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

https://wolfram.com/xid/0dekoi-we18c

Give the result for an arbitrary symbolic direction:

https://wolfram.com/xid/0dekoi-u5c9h

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

https://wolfram.com/xid/0dekoi-frh1ho


https://wolfram.com/xid/0dekoi-bqnb0r

Series expansion for the incomplete gamma function at infinity:

https://wolfram.com/xid/0dekoi-ei0rfx


https://wolfram.com/xid/0dekoi-bnf9an

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

https://wolfram.com/xid/0dekoi-lw3j1b

Gamma can be applied to a power series:

https://wolfram.com/xid/0dekoi-cub5lh

Integral Transforms (4)
Compute the Laplace transform of the incomplete gamma function using LaplaceTransform:

https://wolfram.com/xid/0dekoi-eqbky1

InverseLaplaceTransform of the incomplete gamma function:

https://wolfram.com/xid/0dekoi-c3bnsd

MellinTransform of the incomplete gamma function:

https://wolfram.com/xid/0dekoi-cwgqpu

InverseMellinTransform of the Euler gamma function:

https://wolfram.com/xid/0dekoi-7mn4u

Function Identities and Simplifications (5)

https://wolfram.com/xid/0dekoi-lm2gct

Use FullSimplify to simplify gamma functions:

https://wolfram.com/xid/0dekoi-422vv


https://wolfram.com/xid/0dekoi-bgfkk3

The Euler gamma function basic relation, :

https://wolfram.com/xid/0dekoi-cdx0vd

The Euler gamma function of a double argument, :

https://wolfram.com/xid/0dekoi-fajdil

Relation to the incomplete gamma function:

https://wolfram.com/xid/0dekoi-ge3pv7

Function Representations (5)
Integral representation of the Euler gamma function:

https://wolfram.com/xid/0dekoi-kgkzwa

Integral representation of the incomplete gamma function:

https://wolfram.com/xid/0dekoi-eadyd1

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

https://wolfram.com/xid/0dekoi-bvg3dp


https://wolfram.com/xid/0dekoi-f7v8cg

The incomplete gamma function can be represented as a DifferentialRoot:

https://wolfram.com/xid/0dekoi-czjj10

TraditionalForm formatting:

https://wolfram.com/xid/0dekoi-ppzpg3

Generalizations & Extensions (6)Generalized and extended use cases
Euler Gamma Function (3)
Gamma threads element-wise over lists:

https://wolfram.com/xid/0dekoi

Series expansion at poles:

https://wolfram.com/xid/0dekoi

Expansion at symbolically specified negative integers:

https://wolfram.com/xid/0dekoi

TraditionalForm formatting:

https://wolfram.com/xid/0dekoi-vow6c

Incomplete Gamma Function (1)
Evaluate symbolically at integer and half‐integer orders:

https://wolfram.com/xid/0dekoi


https://wolfram.com/xid/0dekoi

Generalized Incomplete Gamma Function (2)
Evaluate symbolically at integer and half‐integer orders:

https://wolfram.com/xid/0dekoi

Series expansion at a generic point:

https://wolfram.com/xid/0dekoi

Applications (9)Sample problems that can be solved with this function
Plot of the absolute value of Gamma in the complex plane:

https://wolfram.com/xid/0dekoi

Find the asymptotic expansion of ratios of gamma functions:

https://wolfram.com/xid/0dekoi

Volume of an ‐dimensional unit hypersphere:

https://wolfram.com/xid/0dekoi
Low‐dimensional cases:

https://wolfram.com/xid/0dekoi

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

https://wolfram.com/xid/0dekoi

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

https://wolfram.com/xid/0dekoi

CDF of the ‐distribution:

https://wolfram.com/xid/0dekoi-in9jq

Calculate the PDF:

https://wolfram.com/xid/0dekoi

Plot the CDF for different numbers of degrees of freedom:

https://wolfram.com/xid/0dekoi

Compute derivatives of the Gamma function with the BellY polynomial:

https://wolfram.com/xid/0dekoi-lrps17


https://wolfram.com/xid/0dekoi-fpqot

Compute as a limit of Gamma functions at Infinity:

https://wolfram.com/xid/0dekoi-j2ep69

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

https://wolfram.com/xid/0dekoi-f0ryoc


https://wolfram.com/xid/0dekoi-fkufen

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

https://wolfram.com/xid/0dekoi-co4nqb

Represent Zeta in terms of Integrate and the Gamma function:

https://wolfram.com/xid/0dekoi-fbw43s

Properties & Relations (7)Properties of the function, and connections to other functions
Use FullSimplify to simplify gamma functions:

https://wolfram.com/xid/0dekoi-jwowxi

Numerically find a root of a transcendental equation:

https://wolfram.com/xid/0dekoi

Sum expressions involving Gamma:

https://wolfram.com/xid/0dekoi


https://wolfram.com/xid/0dekoi-bwel4d

Generate from integrals, products, and limits:

https://wolfram.com/xid/0dekoi


https://wolfram.com/xid/0dekoi


https://wolfram.com/xid/0dekoi

Obtain Gamma as the solution of a differential equation:

https://wolfram.com/xid/0dekoi

Integrals:

https://wolfram.com/xid/0dekoi

Gamma can be represented as a DifferenceRoot:

https://wolfram.com/xid/0dekoi-pxkma

Possible Issues (2)Common pitfalls and unexpected behavior
Large arguments can give results too large to be computed explicitly:

https://wolfram.com/xid/0dekoi


Machine‐number inputs can give high‐precision results:

https://wolfram.com/xid/0dekoi


https://wolfram.com/xid/0dekoi

Neat Examples (3)Surprising or curious use cases
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).
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.
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
Wolfram Language. (1988). Gamma. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Gamma.html
BibTeX
@misc{reference.wolfram_2025_gamma, author="Wolfram Research", title="{Gamma}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Gamma.html}", note=[Accessed: 06-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_gamma, organization={Wolfram Research}, title={Gamma}, year={2022}, url={https://reference.wolfram.com/language/ref/Gamma.html}, note=[Accessed: 06-May-2025
]}