Gamma
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The gamma function satisfies TemplateBox[{z}, Gamma]=int_0^inftyt^(z-1)e^(-t)dt.
- The incomplete gamma function satisfies TemplateBox[{a, z}, Gamma2]=int_z^inftyt^(a-1)e^(-t)dt.
- 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 all close allBasic 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:
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Compute average-case statistical intervals using Around :
Compute the elementwise values of an array:
Or compute the matrix Gamma function using MatrixFunction :
Specific Values (5)
Singular points of Gamma :
Values at infinity:
Find a local minimum as a root of (dTemplateBox[{x}, Gamma])/(d x)=0:
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 TemplateBox[{z}, Gamma]:
Plot the imaginary part of TemplateBox[{z}, Gamma]:
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 TemplateBox[{x}, Gamma] achieves all nonzero values on the reals:
The incomplete gamma function TemplateBox[{1, x}, Gamma2] achieves all positive real values for real inputs:
On the complexes, however, it achieves all nonzero values:
The incomplete gamma function TemplateBox[{{1, /, 2}, x}, Gamma2] has the restricted range :
The Euler gamma function has the mirror property TemplateBox[{TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox]}, Gamma]=TemplateBox[{TemplateBox[{z}, Gamma]}, Conjugate]:
The complete gamma function TemplateBox[{x}, Gamma] is a meromorphic, nonanalytic function:
TemplateBox[{a, x}, Gamma2] is analytic in for positive integer :
But in general, it is neither an analytic nor a meromorphic function:
TemplateBox[{x}, Gamma] has both singularities and discontinuities on the non-positive integers:
TemplateBox[{x}, Gamma] is neither non-increasing nor non-decreasing:
TemplateBox[{a, x}, Gamma2] is a non-increasing function of when is a positive, odd integer:
But in general, it is neither non-increasing nor non-decreasing:
TemplateBox[{x}, Gamma] is not injective:
TemplateBox[{a, x}, Gamma2] is an injective function of for noninteger :
For integer , it may or may not be injective in :
TemplateBox[{x}, Gamma] is not surjective:
TemplateBox[{a, x}, Gamma2] is also not surjective:
Visualize for :
TemplateBox[{x}, Gamma] is neither non-negative nor non-positive:
TemplateBox[{a, x}, Gamma2] is non-negative for positive odd :
In general, it is neither non-negative nor non-positive:
TemplateBox[{x}, Gamma] is neither convex nor concave:
TemplateBox[{a, x}, Gamma2] 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 (n-1)! = TemplateBox[{n}, Gamma]:
Use FullSimplify to simplify gamma functions:
The Euler gamma function basic relation, TemplateBox[{{z, +, 1}}, Gamma]=z TemplateBox[{z}, Gamma]:
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 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:
Machine‐number inputs can give high‐precision results:
History
Introduced in 1988 (1.0) | Updated in 2021 (13.0) ▪ 2022 (13.1)
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_2025_gamma, author="Wolfram Research", title="{Gamma}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Gamma.html}", note=[Accessed: 24-October-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: 24-October-2025]}