/* REAL: erf(x) = 2x/sqrt(pi) * exp(-x^2) 1F1(1, 3/2, x^2) */
void
acb_hypgeom_erf_1f1b(acb_t res, const acb_t z, slong prec)
{
acb_t a, b, t, w;
acb_init(a);
acb_init(b);
acb_init(t);
acb_init(w);
acb_set_ui(b, 3);
acb_mul_2exp_si(b, b, -1);
acb_mul(w, z, z, prec);
acb_hypgeom_pfq_direct(t, a, 0, b, 1, w, -1, prec);
acb_neg(w, w);
acb_exp(w, w, prec);
acb_mul(t, t, w, prec);
acb_mul(t, t, z, prec);
arb_const_sqrt_pi(acb_realref(w), prec);
acb_div_arb(t, t, acb_realref(w), prec);
acb_mul_2exp_si(res, t, 1);
acb_clear(a);
acb_clear(b);
acb_clear(t);
acb_clear(w);
}
void
acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
{
acb_t t, a, b, w;
acb_init(t);
acb_init(a);
acb_init(b);
acb_init(w);
acb_one(a);
acb_mul_2exp_si(a, a, -1);
acb_add(a, a, nu, prec);
acb_mul_2exp_si(b, nu, 1);
acb_add_ui(b, b, 1, prec);
acb_mul_2exp_si(w, z, 1);
acb_hypgeom_u_asymp(t, a, b, w, -1, prec);
acb_neg(w, z);
acb_exp(w, w, prec);
acb_mul(t, t, w, prec);
acb_mul_2exp_si(w, z, 1);
acb_rsqrt(w, w, prec);
acb_mul(res, t, w, prec);
arb_const_sqrt_pi(acb_realref(w), prec);
acb_mul_arb(res, res, acb_realref(w), prec);
acb_clear(t);
acb_clear(a);
acb_clear(b);
acb_clear(w);
}
void
acb_hypgeom_erf_asymp(acb_t res, const acb_t z, int complementary, slong prec, slong prec2)
{
acb_t a, t, u;
acb_init(a);
acb_init(t);
acb_init(u);
if (!acb_is_exact(z) &&
(arf_cmpabs_ui(arb_midref(acb_realref(z)), prec) < 0) &&
(arf_cmpabs_ui(arb_midref(acb_imagref(z)), prec) < 0))
{
acb_t zmid;
mag_t re_err, im_err;
acb_init(zmid);
mag_init(re_err);
mag_init(im_err);
acb_hypgeom_erf_propagated_error(re_err, im_err, z);
arf_set(arb_midref(acb_realref(zmid)), arb_midref(acb_realref(z)));
arf_set(arb_midref(acb_imagref(zmid)), arb_midref(acb_imagref(z)));
acb_hypgeom_erf_asymp(res, zmid, complementary, prec, prec2);
arb_add_error_mag(acb_realref(res), re_err);
arb_add_error_mag(acb_imagref(res), im_err);
acb_clear(zmid);
mag_clear(re_err);
mag_clear(im_err);
return;
}
acb_one(a);
acb_mul_2exp_si(a, a, -1);
acb_mul(t, z, z, prec2);
acb_hypgeom_u_asymp(u, a, a, t, -1, prec2);
acb_neg(t, t);
acb_exp(t, t, prec2);
acb_mul(u, u, t, prec2);
arb_const_sqrt_pi(acb_realref(t), prec2);
arb_zero(acb_imagref(t));
acb_mul(t, t, z, prec2);
acb_div(u, u, t, prec2);
/* branch cut term: -1 or 1 */
acb_csgn(acb_realref(t), z);
arb_zero(acb_imagref(t));
if (complementary)
{
/* erfc(z) = 1 - erf(z) = u - (sgn - 1) */
acb_sub_ui(t, t, 1, prec);
acb_sub(t, u, t, prec);
}
else
{
/* erf(z) = sgn - u */
acb_sub(t, t, u, prec);
}
if (arb_is_zero(acb_imagref(z)))
{
arb_zero(acb_imagref(t));
}
else if (arb_is_zero(acb_realref(z)))
{
if (complementary)
arb_one(acb_realref(t));
else
arb_zero(acb_realref(t));
}
acb_set(res, t);
acb_clear(a);
acb_clear(t);
acb_clear(u);
}
void
acb_hypgeom_legendre_q(acb_t res, const acb_t n, const acb_t m,
const acb_t z, int type, slong prec)
{
if (type == 0)
{
/* http://functions.wolfram.com/07.11.26.0033.01 */
/* todo: simplify the gamma quotients and the sqrt pi factor... */
acb_t a, b, c, z2, mn, nm, t, u;
acb_init(a);
acb_init(b);
acb_init(c);
acb_init(z2);
acb_init(mn);
acb_init(nm);
acb_init(t);
acb_init(u);
acb_add(mn, m, n, prec); /* mn = m + n */
acb_sub(nm, n, m, prec); /* nm = n - m */
acb_mul(z2, z, z, prec); /* z2 = z^2 */
/* t = 2F1((1-m-n)/2, (n-m)/2+1, 3/2, z^2) */
acb_sub_ui(a, mn, 1, prec);
acb_neg(a, a);
acb_mul_2exp_si(a, a, -1);
acb_mul_2exp_si(b, nm, -1);
acb_add_ui(b, b, 1, prec);
acb_set_ui(c, 3);
acb_mul_2exp_si(c, c, -1);
acb_hypgeom_2f1(t, a, b, c, z2, 0, prec);
/* u = 2F1(-(m+n)/2, (n-m+1)/2, 1/2, z^2) */
acb_neg(a, mn);
acb_mul_2exp_si(a, a, -1);
acb_add_ui(b, nm, 1, prec);
acb_mul_2exp_si(b, b, -1);
acb_one(c);
acb_mul_2exp_si(c, c, -1);
acb_hypgeom_2f1(u, a, b, c, z2, 0, prec);
/* a = cospi((m+n)/2) gamma((m+n)/2+1) rgamma((n-m+1)/2) z */
/* b = sinpi((m+n)/2) gamma((m+n+1)/2) rgamma((n-m)/2+1) / 2 */
acb_mul_2exp_si(a, mn, -1);
acb_sin_cos_pi(b, a, a, prec);
acb_mul_2exp_si(c, mn, -1);
acb_add_ui(c, c, 1, prec);
acb_gamma(c, c, prec);
acb_mul(a, a, c, prec);
acb_add_ui(c, nm, 1, prec);
acb_mul_2exp_si(c, c, -1);
acb_rgamma(c, c, prec);
acb_mul(a, a, c, prec);
acb_mul(a, a, z, prec);
acb_add_ui(c, mn, 1, prec);
acb_mul_2exp_si(c, c, -1);
acb_gamma(c, c, prec);
acb_mul(b, b, c, prec);
acb_mul_2exp_si(c, nm, -1);
acb_add_ui(c, c, 1, prec);
acb_rgamma(c, c, prec);
acb_mul(b, b, c, prec);
acb_mul_2exp_si(b, b, -1);
/* at - bu */
acb_mul(t, t, a, prec);
acb_mul(u, u, b, prec);
acb_sub(t, t, u, prec);
/* prefactor sqrt(pi) 2^m (1-z^2)^(-m/2) */
if (!acb_is_zero(m))
{
acb_sub_ui(u, z2, 1, prec);
acb_neg(u, u);
acb_neg(c, m);
acb_mul_2exp_si(c, c, -1);
acb_pow(u, u, c, prec);
acb_set_ui(c, 2);
acb_pow(c, c, m, prec);
acb_mul(u, u, c, prec);
acb_mul(t, t, u, prec);
}
arb_const_sqrt_pi(acb_realref(u), prec);
acb_mul_arb(t, t, acb_realref(u), prec);
acb_set(res, t);
acb_clear(a);
acb_clear(b);
acb_clear(c);
acb_clear(z2);
acb_clear(mn);
acb_clear(nm);
acb_clear(t);
acb_clear(u);
}
else if (type == 1)
{
if ((arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), -2) < 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), -2) < 0) ||
!_acb_hypgeom_legendre_q_single_valid(z))
{
_acb_hypgeom_legendre_q_double(res, n, m, z, prec);
}
else
{
_acb_hypgeom_legendre_q_single(res, n, m, z, prec);
}
}
else
{
flint_printf("unsupported 'type' %d for legendre q\n", type);
abort();
}
}
void
_acb_hypgeom_legendre_q_single(acb_t res, const acb_t n, const acb_t m,
const acb_t z, slong prec)
{
acb_t a, b, c, z2, t, u;
acb_init(a);
acb_init(b);
acb_init(c);
acb_init(z2);
acb_init(t);
acb_init(u);
/* invalid in (-1,0) */
if (!_acb_hypgeom_legendre_q_single_valid(z))
{
acb_indeterminate(res);
return;
}
acb_pow_si(z2, z, -2, prec); /* z2 = 1/z^2 */
/* t = 2F1r((m+n+1)/2, (m+n)/2+1, n+3/2, 1/z^2) */
acb_add(b, m, n, prec);
acb_add_ui(a, b, 1, prec);
acb_mul_2exp_si(a, a, -1);
acb_mul_2exp_si(b, b, -1);
acb_add_ui(b, b, 1, prec);
acb_set_ui(c, 3);
acb_mul_2exp_si(c, c, -1);
acb_add(c, c, n, prec);
acb_hypgeom_2f1(t, a, b, c, z2, 1, prec);
/* prefactor sqrt(pi) 2^-n (z+1)^(m/2) (z-1)^(m/2) exp(i pi m) */
/* (1/2) gamma(m+n+1) z^(-m-n-1) */
if (!acb_is_zero(m))
{
acb_add_ui(z2, z, 1, prec);
acb_mul_2exp_si(c, m, -1);
acb_pow(z2, z2, c, prec);
acb_mul(t, t, z2, prec);
acb_sub_ui(z2, z, 1, prec);
acb_mul_2exp_si(c, m, -1);
acb_pow(z2, z2, c, prec);
acb_mul(t, t, z2, prec);
acb_exp_pi_i(z2, m, prec);
acb_mul(t, t, z2, prec);
}
acb_set_ui(z2, 2);
acb_neg(c, n);
acb_pow(z2, z2, c, prec);
acb_mul(t, t, z2, prec);
acb_add(c, m, n, prec);
acb_add_ui(c, c, 1, prec);
acb_gamma(z2, c, prec);
acb_mul(t, t, z2, prec);
acb_neg(c, c);
acb_pow(z2, z, c, prec);
acb_mul(t, t, z2, prec);
acb_mul_2exp_si(t, t, -1);
arb_const_sqrt_pi(acb_realref(u), prec);
acb_mul_arb(t, t, acb_realref(u), prec);
acb_set(res, t);
acb_clear(a);
acb_clear(b);
acb_clear(c);
acb_clear(z2);
acb_clear(t);
acb_clear(u);
}