GammaRegularized

GammaRegularized[a,z]

is the regularized incomplete gamma function TemplateBox[{a, z}, GammaRegularized].

Details

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (41)

Numerical Evaluation  (6)

Evaluate numerically:

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:

GammaRegularized can be used with Interval and CenteredInterval objects:

Specific Values  (5)

Values at specific points:

Values at infinity:

Evaluate at integer and halfinteger arguments:

The generalized regularized incomplete gamma function at integer and halfinteger arguments:

Find the zero of TemplateBox[{2, x}, GammaRegularized]:

Visualization  (3)

Plot the regularized gamma function for integer arguments:

Plot the regularized gamma function for half-integer arguments:

Plot the real part of TemplateBox[{3, z}, GammaRegularized]:

Plot the imaginary part of TemplateBox[{3, z}, GammaRegularized]:

Function Properties  (9)

Real domain of TemplateBox[{a, x}, GammaRegularized]:

Complex domain:

The regularized incomplete gamma function TemplateBox[{1, x}, GammaRegularized] achieves all positive real values for real inputs:

The range for complex values:

TemplateBox[{{1, /, 2}, x}, GammaRegularized] has the restricted range :

TemplateBox[{a, x}, GammaRegularized] is an analytic function of for positive integer :

For other values of , it may or may not be analytic:

When it is not analytic, it is also not meromorphic:

TemplateBox[{1, x}, GammaRegularized] has no singularities or discontinuities:

TemplateBox[{{1, /, 2}, x}, GammaRegularized] has singularities and discontinuities for :

TemplateBox[{a, x}, GammaRegularized] is a non-increasing function of when is a positive, odd integer:

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

TemplateBox[{a, x}, GammaRegularized] is an injective function of for noninteger :

For other values of , it may or may not be injective in :

TemplateBox[{a, x}, GammaRegularized] is not a surjective function of for most values of :

Visualize for :

TemplateBox[{a, x}, GammaRegularized] is non-negative for positive odd :

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

TemplateBox[{1, x}, GammaRegularized] is convex:

TemplateBox[{{-, {1, /, 2}}, x}, GammaRegularized] is concave on its real domain:

TemplateBox[{2, x}, GammaRegularized] is neither convex nor concave:

Differentiation  (2)

First derivative of the regularized incomplete gamma function:

Higher derivatives:

Plot higher derivatives for integer and half-integer :

Integration  (3)

Indefinite integral of the regularized incomplete gamma function:

Definite integral int_0^inftyTemplateBox[{a, x}, GammaRegularized]dx:

More integrals:

Series Expansions  (4)

Series expansion for the regularized incomplete gamma function:

Plot the first three approximations for TemplateBox[{1, x}, GammaRegularized] around :

Series expansion at infinity:

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)

Compute the Laplace transform using LaplaceTransform:

MellinTransform:

Function Identities and Simplifications  (3)

FunctionExpand regularized gamma functions through ordinary gamma functions:

Use FullSimplify to simplify regularized gamma functions:

Recurrence identity:

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 halfinteger arguments:

Infinite arguments give symbolic results:

GammaRegularized threads elementwise over lists:

Generalized Regularized Incomplete Gamma Function  (1)

Evaluate at integer and halfinteger arguments:

Applications  (5)

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 RiemannLiouville integral:

Fractional derivative/integral of integer orders:

Plot fractional derivative/integral:

A liquid crystal display (LCD) has 1920×1080 pixels. A display is accepted if it has 15 or fewer faulty pixels. The probability that a pixel is faulty from production is . Find the proportion of displays that are accepted:

Find the pixel failure rate required to produce 4000×2000 pixel displays and still have an acceptance rate of at least 90%:

Plot the acceptance rate as a function of the pixel failure rate:

Find the maximal acceptable pixel failure rate:

Check the result:

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:

Machinenumber inputs can give highprecision 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:

Wolfram Research (1991), GammaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaRegularized.html (updated 2022).

Text

Wolfram Research (1991), GammaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaRegularized.html (updated 2022).

CMS

Wolfram Language. 1991. "GammaRegularized." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. 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

BibTeX

@misc{reference.wolfram_2023_gammaregularized, author="Wolfram Research", title="{GammaRegularized}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/GammaRegularized.html}", note=[Accessed: 18-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_gammaregularized, organization={Wolfram Research}, title={GammaRegularized}, year={2022}, url={https://reference.wolfram.com/language/ref/GammaRegularized.html}, note=[Accessed: 18-March-2024 ]}