//================================================================
/**
 * @packageDocumentation
 * @module std
 */
//================================================================
import { MathUtil } from "../../internal/numeric/MathUtil";
import { tgamma } from "./gamma";

/**
 * Riemann zeta function.
 *
 * @reference http://en.cppreference.com/w/cpp/numeric/special_math/riemann_zeta
 */
export function riemann_zeta(arg: number): number {
    if (arg < 0) return _Negative(arg);
    else if (arg === 0) return -0.5;
    else if (arg < 1) return _Fractional(arg);
    else if (arg === 1) return Infinity;
    else return _Positive(arg);
}

function _Negative(arg: number): number {
    return (
        Math.pow(2, arg) *
        Math.pow(Math.PI, arg - 1) *
        Math.sin((Math.PI * arg) / 2) *
        tgamma(1 - arg) *
        riemann_zeta(1 - arg)
    );
}

function _Fractional(arg: number): number {
    const divider: number = 1 - Math.pow(2, 1 - arg);
    const sigma: number = MathUtil.sigma(
        function (n: number): number {
            return Math.pow(-1, n - 1) * Math.pow(n, -arg);
        },
        1,
        INFINITY,
    );

    return sigma / divider;
}

function _Positive(arg: number): number {
    return MathUtil.sigma(
        function (n: number): number {
            return Math.pow(n, -arg);
        },
        1,
        INFINITY,
    );
}

const INFINITY = 100 * 1000;