# GammaRegularized

GammaRegularized[a,z]

is the regularized incomplete gamma function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• In nonsingular cases, .
• GammaRegularized[a,z0,z1] is the generalized regularized incomplete gamma function, defined in nonsingular 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

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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(35)

### Numerical Evaluation(5)

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:

### 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(4)

Real domain of :

Complex domain:

The regularized incomplete gamma function for an integer argument takes all real positive values:

The range for complex values:

The regularized incomplete gamma function for takes all real values from the interval [ ]:

The regularized incomplete gamma function for takes all real negative values:

### 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(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: 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:

Introduced in 1991
(2.0)