About stdlib...

We believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.

The library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.

When you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.

To join us in bringing numerical computing to the web, get started by checking us out on GitHub, and please consider financially supporting stdlib. We greatly appreciate your continued support!

gammainc

Incomplete gamma function.

Evaluates the regularized lower incomplete gamma function:

$$P( x, a ) = \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} \; dt$$

The function can also be used to evaluate the regularized upper incomplete gamma function, which is defined as follows:

$$Q( x, a ) = \frac{\Gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t} \; dt$$

The two functions have the relationship Q(x,a) = 1 - P(x,a).

In addition, this routine can be used to evaluate the unregularized gamma functions. The range of above functions is [0,1], which is not the case for the unregularized versions. The unregularized lower incomplete gamma function is defined as

$$\gamma(a,x) = \int_0^x t^{a-1} e^{-t} \; dt$$

and the upper unregularized incomplete gamma function is

$$\Gamma(a,x)= \int_x^\infty t^{a-1} e^{-t} \; dt$$

The relationship between the two functions is Γ(a,x) = γ(a,x) + Γ(a).

Installation

npm install @stdlib/math-base-special-gammainc

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var gammainc = require( '@stdlib/math-base-special-gammainc' );

gammainc( x, s[, regularized[, upper ]] )

By default, evaluates the regularized lower incomplete gamma function for inputs x and s. The third and fourth parameters of the function can be used to specify whether instead to evaluate the non-regularized and/or upper incomplete gamma functions, respectively.

var y = gammainc( 6.0, 2.0 );
// returns ~0.9826

y = gammainc( 1.0, 2.0, true, true );
// returns ~0.7358

y = gammainc( 7.0, 5.0 );
// returns ~0.8270

y = gammainc( 7.0, 5.0, false );
// returns ~19.8482

If provided NaN as any argument, the function returns NaN.

var y = gammainc( NaN, 1.0 );
// returns NaN

y = gammainc( 1.0, NaN );
// returns NaN

If provided a negative x, the function returns NaN.

var y = gammainc( -2.0, 2.0 );
// returns NaN

If provided a nonpositive s, the function returns NaN.

var y = gammainc( 2.0, -1.0 );
// returns NaN

y = gammainc( 2.0, 0.0 );
// returns NaN

Examples

var randu = require( '@stdlib/random-base-randu' );
var gammainc = require( '@stdlib/math-base-special-gammainc' );

var i;
var x;
var s;

for ( i = 0; i < 100; i++ ) {
    x = randu() * 10.0;
    s = randu() * 10.0;
    console.log( 'x: %d, \t s: %d, \t f(x,s): %d', x.toFixed( 4 ), s.toFixed( 4 ), gammainc( x, s ).toFixed( 4 ) );
}

---

See Also

• @stdlib/math-base/special/gamma: gamma function.
• @stdlib/math-base/special/gamma1pm1: compute gamma(x+1) - 1.
• @stdlib/math-base/special/gammaincinv: inverse incomplete gamma function.
• @stdlib/math-base/special/gammaln: natural logarithm of the gamma function.

---

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

---

Copyright

Copyright © 2016-2024. The Stdlib Authors.