Inverse of the regularized incomplete gamma function

Inverse of the regularized incomplete gamma function—Wolfram Documentation

Function Identities and Simplifications

Generalizations & Extensions

Details

Examples

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Basic Examples (3)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at x=-1:

Scope (30)

Numerical Evaluation (5)

Evaluate numerically to high precision:

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

Evaluate InverseGammaRegularized efficiently at high precision:

Evaluate the three-argument generalized case:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix InverseGammaRegularized function using MatrixFunction:

Specific Values (3)

Values at fixed points:

Find the zero of :

Real domain of :

Visualization (2)

Plot the inverse of the regularized gamma function for integer arguments:

Plot the real part of :

Function Properties (8)

Differentiation (3)

First derivative of the inverse of the regularized incomplete gamma function:

Higher derivatives:

First derivative of the inverse of the generalized regularized incomplete gamma function:

Integration (2)

Indefinite integral of the inverse regularized incomplete gamma function:

Definite integral $int_0^1TemplateBox[{1, s}, InverseGammaRegularized]ds$ :

Series Expansions (3)

Function Identities and Simplifications (2)

Other Features (2)

Generalizations & Extensions (1)

Applications (2)

Model the PDF of the gamma distribution through uniformly distributed random numbers:

Compare binned modeled distribution with exact distribution:

Quartiles for a derived distribution:

Data distribution:

Properties & Relations (2)

Possible Issues (2)

History

Introduced in 1996 (3.0)

Wolfram Research (1996), InverseGammaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseGammaRegularized.html.

Text

Wolfram Research (1996), InverseGammaRegularized, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseGammaRegularized.html.

CMS

Wolfram Language. 1996. "InverseGammaRegularized." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseGammaRegularized.html.

APA

Wolfram Language. (1996). InverseGammaRegularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseGammaRegularized.html

BibTeX

@misc{reference.wolfram_2025_inversegammaregularized, author="Wolfram Research", title="{InverseGammaRegularized}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/InverseGammaRegularized.html}", note=[Accessed: 22-February-2026]}

BibLaTeX

@online{reference.wolfram_2025_inversegammaregularized, organization={Wolfram Research}, title={InverseGammaRegularized}, year={1996}, url={https://reference.wolfram.com/language/ref/InverseGammaRegularized.html}, note=[Accessed: 22-February-2026]}