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/*
Copyright (C) 2017 Fredrik Johansson
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "acb.h"
static int
_acb_lambertw_check_branch(const acb_t w, const fmpz_t k, slong prec)
{
arb_t t, u, v, a, b;
int res = 0;
arb_init(t);
arb_init(u);
arb_init(v);
arb_init(a);
arb_init(b);
/* t = x sinc(y), v = -cos(y) */
if (arb_is_exact(acb_imagref(w)))
{
if (arb_is_zero(acb_imagref(w)))
{
arb_one(t);
arb_one(v);
}
else
{
arb_sin_cos(t, v, acb_imagref(w), prec);
arb_div(t, t, acb_imagref(w), prec);
}
}
else
{
arb_sinc(t, acb_imagref(w), prec);
arb_cos(v, acb_imagref(w), prec);
}
arb_mul(t, t, acb_realref(w), prec);
arb_neg(v, v);
/* u = y / pi, with conjugate relation for k */
arb_const_pi(u, prec);
arb_div(u, acb_imagref(w), u, prec);
if (fmpz_sgn(k) < 0)
arb_neg(u, u);
if (fmpz_is_zero(k))
{
/* -1 < u < 1 and t > v */
arb_set_si(a, -1);
arb_set_si(b, 1);
if (arb_gt(u, a) && arb_lt(u, b) && arb_gt(t, v))
{
res = 1;
}
}
else
{
arb_set_fmpz(a, k);
arb_abs(a, a);
arb_mul_2exp_si(a, a, 1);
arb_add_ui(b, a, 1, prec);
arb_sub_ui(a, a, 2, prec);
/* if u > 2|k|-2 and u < 2|k|+1 */
if (arb_gt(u, a) && arb_lt(u, b))
{
arb_add_ui(a, a, 1, prec);
arb_sub_ui(b, b, 1, prec);
/* if u > 2|k|-1 and u < 2|k| */
if (arb_gt(u, a) && arb_lt(u, b))
{
res = 1;
}
else if (arb_lt(u, b) && arb_lt(t, v)) /* u < 2|k| and t < v */
{
res = 1;
}
else if (arb_gt(u, a) && arb_gt(t, v)) /* u > 2|k|-1 and t > v */
{
res = 1;
}
}
}
arb_clear(t);
arb_clear(u);
arb_clear(v);
arb_clear(a);
arb_clear(b);
return res;
}
int
acb_lambertw_check_branch(const acb_t w, const fmpz_t k, slong prec)
{
if (prec > 64 && _acb_lambertw_check_branch(w, k, 32))
return 1;
return _acb_lambertw_check_branch(w, k, prec);
}