gammainc

Incomplete gamma function.

Evaluates the regularized lower incomplete gamma function:

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

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

and the upper unregularized incomplete gamma function is

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

Installation

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

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.

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