@stdlib/math-base-special-gammainc/lib/finite_half_gamma_q.js

Incomplete gamma function.

/**
* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
*    http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*
* ## Notice
*
* The original C++ code and copyright notice are from the [Boost library]{@link https://www.boost.org/doc/libs/1_88_0/boost/math/special_functions/gamma.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
* (C) Copyright Paul A. Bristow 2007.
*
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
* ```
*/

'use strict';

// MODULES //

var erfc = require( '@stdlib/math-base-special-erfc' );
var sqrt = require( '@stdlib/math-base-special-sqrt' );
var exp = require( '@stdlib/math-base-special-exp' );
var PI = require( '@stdlib/constants-float64-pi' );


// MAIN //

/**
* Calculates normalized Q when a is a half-integer.
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @returns {number} upper gamma fraction
*/
function finiteHalfGammaQ( a, x ) {
	var half;
	var term;
	var sum;
	var e;
	var n;

	e = erfc( sqrt(x) );
	if ( e !== 0 && a > 1.0 ) {
		term = exp( -x ) / sqrt( PI * x );
		term *= x;
		half = 0.5;
		term /= half;
		sum = term;
		for ( n = 2; n < a; ++n ) {
			term /= n - half;
			term *= x;
			sum += term;
		}
		e += sum;
	}
	return e;
}


// EXPORTS //

module.exports = finiteHalfGammaQ;