# GammaRegularized

GammaRegularized[a,z]

is the regularized incomplete gamma function .

# Examples

open allclose all

## 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 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 :

### Visualization(3)

Plot the regularized gamma function for integer arguments:

Plot the regularized gamma function for half-integer arguments:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Real domain of :

Complex domain:

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

The range for complex values:

has the restricted range :

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:

has no singularities or discontinuities:

has singularities and discontinuities for :

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

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

is an injective function of for noninteger :

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

is not a surjective function of for most values of :

Visualize for :

is non-negative for positive odd :

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

is convex:

is concave on its real domain:

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 :

More integrals:

### Series Expansions(4)

Series expansion for the regularized incomplete gamma function:

Plot the first three approximations for 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:

### 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:

## Generalizations & Extensions(4)

### Regularized Incomplete Gamma Function(3)

Evaluate at integer and halfinteger arguments:

Infinite arguments give symbolic results:

### 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_2024_gammaregularized, author="Wolfram Research", title="{GammaRegularized}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/GammaRegularized.html}", note=[Accessed: 15-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_gammaregularized, organization={Wolfram Research}, title={GammaRegularized}, year={2022}, url={https://reference.wolfram.com/language/ref/GammaRegularized.html}, note=[Accessed: 15-July-2024 ]}