GammaRegularized
GammaRegularized[a,z]
is the regularized incomplete gamma function .
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- In non‐singular cases,
.
- GammaRegularized[a,z0,z1] is the generalized regularized incomplete gamma function, defined in non‐singular cases as Gamma[a,z0,z1]/Gamma[a].
- Note that the arguments in GammaRegularized are arranged differently from those in BetaRegularized.
- For certain special arguments, GammaRegularized automatically evaluates to exact values.
- GammaRegularized can be evaluated to arbitrary numerical precision.
- GammaRegularized automatically threads over lists.
Examples
open allclose allBasic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Scope (35)
Numerical Evaluation (5)
Evaluate numerically to high precision:
The precision of the output tracks the precision of the input:
Evaluate numerically for complex arguments:
Evaluate GammaRegularized efficiently at high precision:
GammaRegularized threads elementwise over lists:
Specific Values (5)
Visualization (3)
Function Properties (4)
Differentiation (2)
Integration (3)
Series Expansions (4)
Series expansion for the regularized incomplete gamma function:
Plot the first three approximations for around
:
Give the result for an arbitrary symbolic direction:
Expansions of the generalized regularized incomplete gamma function at a generic point:
GammaRegularized can be applied to a power series:
Integral Transforms (2)
Function Identities and Simplifications (3)
FunctionExpand regularized gamma functions through ordinary gamma functions:
Use FullSimplify to simplify regularized gamma functions:
Function Representations (4)
Integral representation of the regularized incomplete gamma:
Representation in terms of MeijerG:
GammaRegularized can be represented as a DifferentialRoot:
TraditionalForm formatting:
Generalizations & Extensions (4)
Regularized Incomplete Gamma Function (3)
Evaluate at integer and half‐integer arguments:
Infinite arguments give symbolic results:
GammaRegularized threads element‐wise over lists:
Generalized Regularized Incomplete Gamma Function (1)
Evaluate at integer and half‐integer arguments:
Applications (4)
Plot of the real part of GammaRegularized over the complex plane:
CDF of the distribution:
Calculate PDF:
Plot the CDFs for various degrees of freedom:
CDF of the gamma distribution:
Calculate PDF:
Plot the CDFs for various parameters:
Fractional derivatives/integrals of the exponential function:
Check that this is the defining Liouville integral:
Fractional derivative/integral of integer orders:
Plot fractional derivative/integral:
Properties & Relations (4)
Use FullSimplify to simplify regularized gamma functions:
Use FunctionExpand to express regularized gamma functions through ordinary gamma functions:
Solve a transcendental equation:

Numerically find a root of a transcendental equation:
Possible Issues (3)
Large arguments can underflow and produce a machine zero:

Machine‐number inputs can give high‐precision results:
Gamma rather than GammaRegularized is usually generated in computations:
Regularized gamma functions are typically not generated by FullSimplify:
Neat Examples (3)
Nest GammaRegularized over the complex plane:
Plot GammaRegularized at infinity:
Riemann surface of the incomplete regularized gamma function:
Text
Wolfram Research (1991), GammaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaRegularized.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1991. "GammaRegularized." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GammaRegularized.html.
APA
Wolfram Language. (1991). GammaRegularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GammaRegularized.html