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).

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 uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var gammainc = require( '@stdlib/math/base/special/gammainc' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, 0.0, 10.0, opts );
var s = uniform( 100, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, \t s: %0.4f, \t f(x,s): %0.4f', x, s, gammainc );

---

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.