Built-in Mathematica Symbol

GammaRegularized

GammaRegularized[a, z]
is the regularized incomplete gamma function

.

MORE INFORMATION

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 (3)

Evaluate numerically:

Scope (4)

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate numerically for complex arguments:

TraditionalForm formatting:

Generalizations & Extensions (7)

Evaluate at integer and half-integer arguments:

Infinite arguments give symbolic results:

GammaRegularized threads element-wise over lists:

GammaRegularized can be applied to a power series:

Series expansion at infinity:

Give the result for an arbitrary symbolic direction:

Evaluate at integer and half-integer arguments:

Series expansion at generic point:

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 through ordinary gamma functions:

Solve a transcendental equation:

Numerically find a root of a transcendental equation:

Possible Issues (3)

Large arguments can give results too small to be computed explicitly:

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: