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  (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:

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 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, {x, +, {ⅈ, , y}}}, GammaRegularized]:

Plot the imaginary part of TemplateBox[{3, {x, +, {ⅈ, , y}}}, GammaRegularized]:

Function Properties  (4)

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

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 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 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:

Introduced in 1991
 (2.0)