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/*
Copyright (C) 2014 Fredrik Johansson
This file is part of Arb.
Arb 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 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_elliptic.h"
void
acb_elliptic_k_jet(acb_ptr w, const acb_t m, slong len, slong prec)
{
acb_t t, u, msub1m, m2sub1;
slong k, n;
if (len < 1)
return;
if (len == 1)
{
acb_elliptic_k(w, m, prec);
return;
}
if (acb_is_zero(m))
{
acb_const_pi(w, prec);
acb_mul_2exp_si(w, w, -1);
for (k = 1; k < len; k++)
{
acb_mul_ui(w + k, w + k - 1, (2 * k - 1) * (2 * k - 1), prec);
acb_div_ui(w + k, w + k, 4 * k * k, prec);
}
return;
}
acb_init(t);
acb_init(u);
acb_init(msub1m);
acb_init(m2sub1);
acb_sub_ui(msub1m, m, 1, prec);
acb_neg(t, msub1m);
acb_sqrt(t, t, prec);
acb_mul(msub1m, msub1m, m, prec);
acb_mul_2exp_si(m2sub1, m, 1);
acb_sub_ui(m2sub1, m2sub1, 1, prec);
acb_agm1_cpx(w, t, 2, prec);
/* pi M'(t) / (4 t M(t)^2) */
acb_mul(u, w, w, prec);
acb_mul(t, t, u, prec);
acb_div(w + 1, w + 1, t, prec);
acb_const_pi(u, prec);
acb_mul(w + 1, w + 1, u, prec);
acb_mul_2exp_si(w + 1, w + 1, -2);
/* pi / (2 M(t)) */
acb_const_pi(u, prec);
acb_div(w, u, w, prec);
acb_mul_2exp_si(w, w, -1);
acb_inv(t, msub1m, prec);
for (k = 2; k < len; k++)
{
n = k - 2;
acb_mul_ui(w + k, w + n, (2 * n + 1) * (2 * n + 1), prec);
acb_mul(u, w + n + 1, m2sub1, prec);
acb_addmul_ui(w + k, u, (n + 1) * (n + 1) * 4, prec);
acb_mul(w + k, w + k, t, prec);
acb_div_ui(w + k, w + k, 4 * (n + 1) * (n + 2), prec);
acb_neg(w + k, w + k);
}
acb_clear(t);
acb_clear(u);
acb_clear(msub1m);
acb_clear(m2sub1);
}
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