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:

GammaRegularized can be used with Interval and CenteredInterval objects:

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 :

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 :

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:

First derivative of the regularized incomplete gamma function:

Higher derivatives:

Plot higher derivatives for integer and half-integer :

Indefinite integral of the regularized incomplete gamma function:

Definite integral :

More integrals:

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:

Compute the Laplace transform using LaplaceTransform:

MellinTransform:

FunctionExpand regularized gamma functions through ordinary gamma functions:

Use FullSimplify to simplify regularized gamma functions:

Recurrence identity:

Integral representation of the regularized incomplete gamma:

Representation in terms of MeijerG:

GammaRegularized can be represented as a DifferentialRoot:

TraditionalForm formatting:

Evaluate at integer and half‐integer arguments:

Infinite arguments give symbolic results:

GammaRegularized threads element‐wise over lists:

Evaluate at integer and half‐integer arguments: