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

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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 (40)

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:

GammaRegularized threads elementwise over lists:

Specific Values (5)

Values at specific points:

Values at infinity:

Evaluate at integer and half‐integer arguments:

The generalized regularized incomplete gamma function at integer and half‐integer 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 $int_0^inftyTemplateBox[{a, x}, GammaRegularized]dx$ :

More integrals:

Series Expansions (4)

Series expansion for the regularized incomplete gamma function:

Plot the first three approximations for 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 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:

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GammaRegularized

Details

Examples

Basic Examples (5)