/* 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);
}
static void
_acb_hypgeom_m_1f1(acb_t res, const acb_t a, const acb_t b, const acb_t z, long prec)
{
if (acb_is_one(a))
{
acb_hypgeom_pfq_direct(res, NULL, 0, b, 1, z, -1, prec);
}
else
{
acb_struct c[3];
c[0] = *a;
c[1] = *b;
acb_init(c + 2);
acb_one(c + 2);
acb_hypgeom_pfq_direct(res, c, 1, c + 1, 2, z, -1, prec);
acb_clear(c + 2);
}
}
void
acb_hypgeom_bessel_j_0f1(acb_t res, const acb_t nu, const acb_t z, long prec)
{
acb_struct b[2];
acb_t w, c, t;
if (acb_is_int(nu) && arb_is_negative(acb_realref(nu)))
{
acb_init(t);
acb_neg(t, nu);
acb_hypgeom_bessel_j_0f1(res, t, z, prec);
acb_mul_2exp_si(t, t, -1);
if (!acb_is_int(t))
acb_neg(res, res);
acb_clear(t);
return;
}
acb_init(b + 0);
acb_init(b + 1);
acb_init(w);
acb_init(c);
acb_init(t);
acb_add_ui(b + 0, nu, 1, prec);
acb_one(b + 1);
/* (z/2)^nu / gamma(nu+1) */
acb_mul_2exp_si(c, z, -1);
acb_pow(c, c, nu, prec);
acb_rgamma(t, b + 0, prec);
acb_mul(c, t, c, prec);
/* -z^2/4 */
acb_mul(w, z, z, prec);
acb_mul_2exp_si(w, w, -2);
acb_neg(w, w);
acb_hypgeom_pfq_direct(t, NULL, 0, b, 2, w, -1, prec);
acb_mul(res, t, c, prec);
acb_clear(b + 0);
acb_clear(b + 1);
acb_clear(w);
acb_clear(c);
acb_clear(t);
}
void
acb_hypgeom_ci_2f3(acb_t res, const acb_t z, slong prec)
{
acb_t a, t, u;
acb_struct b[3];
acb_init(a);
acb_init(b);
acb_init(b + 1);
acb_init(b + 2);
acb_init(t);
acb_init(u);
acb_one(a);
acb_set_ui(b, 2);
acb_set(b + 1, b);
acb_set_ui(b + 2, 3);
acb_mul_2exp_si(b + 2, b + 2, -1);
acb_mul(t, z, z, prec);
acb_mul_2exp_si(t, t, -2);
acb_neg(t, t);
acb_hypgeom_pfq_direct(u, a, 1, b, 3, t, -1, prec);
acb_mul(u, u, t, prec);
acb_log(t, z, prec);
acb_add(u, u, t, prec);
arb_const_euler(acb_realref(t), prec);
arb_add(acb_realref(u), acb_realref(u), acb_realref(t), prec);
acb_swap(res, u);
acb_clear(a);
acb_clear(b);
acb_clear(b + 1);
acb_clear(b + 2);
acb_clear(t);
acb_clear(u);
}
/* IMAG: erf(z) = 2z/sqrt(pi) * 1F1(1/2, 3/2, -z^2) */
void
acb_hypgeom_erf_1f1a(acb_t res, const acb_t z, slong prec)
{
acb_t a, t, w;
acb_struct b[2];
acb_init(a);
acb_init(b);
acb_init(b + 1);
acb_init(t);
acb_init(w);
acb_one(a);
acb_mul_2exp_si(a, a, -1);
acb_set_ui(b, 3);
acb_mul_2exp_si(b, b, -1);
acb_one(b + 1);
acb_mul(w, z, z, prec);
acb_neg(w, w);
acb_hypgeom_pfq_direct(t, a, 1, b, 2, w, -1, prec);
acb_const_pi(w, prec);
acb_rsqrt(w, w, prec);
acb_mul(t, t, w, prec);
acb_mul(t, t, z, prec);
acb_mul_2exp_si(res, t, 1);
acb_clear(a);
acb_clear(b);
acb_clear(b + 1);
acb_clear(t);
acb_clear(w);
}
void
acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
{
if (acb_is_int(nu))
{
acb_poly_t nux, zx, rx;
acb_poly_init(nux);
acb_poly_init(zx);
acb_poly_init(rx);
acb_poly_set_coeff_acb(nux, 0, nu);
acb_poly_set_coeff_si(nux, 1, 1);
acb_poly_set_acb(zx, z);
acb_hypgeom_bessel_k_0f1_series(rx, nux, zx, 1, prec);
acb_poly_get_coeff_acb(res, rx, 0);
acb_poly_clear(nux);
acb_poly_clear(zx);
acb_poly_clear(rx);
}
else
{
acb_t t, u, v, w;
acb_struct b[2];
acb_init(t);
acb_init(u);
acb_init(v);
acb_init(w);
acb_init(b + 0);
acb_init(b + 1);
/* u = 0F1(1+nu), v = 0F1(1-nu) */
acb_mul(t, z, z, prec);
acb_mul_2exp_si(t, t, -2);
acb_add_ui(b, nu, 1, prec);
acb_one(b + 1);
acb_hypgeom_pfq_direct(u, NULL, 0, b, 2, t, -1, prec);
acb_sub_ui(b, nu, 1, prec);
acb_neg(b, b);
acb_hypgeom_pfq_direct(v, NULL, 0, b, 2, t, -1, prec);
/* v = v * gamma(nu) / (z/2)^nu */
acb_mul_2exp_si(t, z, -1);
acb_pow(t, t, nu, prec);
acb_gamma(w, nu, prec);
acb_mul(v, v, w, prec);
acb_div(v, v, t, prec);
/* u = u * t * pi / (gamma(nu) * nu * sin(pi nu)) */
acb_mul(u, u, t, prec);
acb_const_pi(t, prec);
acb_mul(u, u, t, prec);
acb_sin_pi(t, nu, prec);
acb_mul(t, t, w, prec);
acb_mul(t, t, nu, prec);
acb_div(u, u, t, prec);
acb_sub(res, v, u, prec);
acb_mul_2exp_si(res, res, -1);
acb_clear(t);
acb_clear(u);
acb_clear(v);
acb_clear(w);
acb_clear(b + 0);
acb_clear(b + 1);
}
}
void
_acb_hypgeom_m_1f1(acb_t res, const acb_t a, const acb_t b, const acb_t z,
int regularized, slong prec, slong gamma_prec, int kummer)
{
if (regularized)
{
/* Remove singularity */
if (acb_is_int(b) && arb_is_nonpositive(acb_realref(b)) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(b)), 30) < 0)
{
acb_t c, d, t, u;
slong n;
n = arf_get_si(arb_midref(acb_realref(b)), ARF_RND_DOWN);
acb_init(c);
acb_init(d);
acb_init(t);
acb_init(u);
acb_sub(c, a, b, prec);
acb_add_ui(c, c, 1, prec);
acb_neg(d, b);
acb_add_ui(d, d, 2, prec);
_acb_hypgeom_m_1f1(t, c, d, z, 0, prec, gamma_prec, kummer);
acb_pow_ui(u, z, 1 - n, prec);
acb_mul(t, t, u, prec);
acb_rising_ui(u, a, 1 - n, prec);
acb_mul(t, t, u, prec);
arb_fac_ui(acb_realref(u), 1 - n, prec);
acb_div_arb(res, t, acb_realref(u), prec);
acb_clear(c);
acb_clear(d);
acb_clear(t);
acb_clear(u);
}
else
{
acb_t t;
acb_init(t);
acb_rgamma(t, b, gamma_prec);
_acb_hypgeom_m_1f1(res, a, b, z, 0, prec, gamma_prec, kummer);
acb_mul(res, res, t, prec);
acb_clear(t);
}
return;
}
/* Kummer's transformation */
if (kummer)
{
acb_t u, v;
acb_init(u);
acb_init(v);
acb_sub(u, b, a, prec);
acb_neg(v, z);
_acb_hypgeom_m_1f1(u, u, b, v, regularized, prec, gamma_prec, 0);
acb_exp(v, z, prec);
acb_mul(res, u, v, prec);
acb_clear(u);
acb_clear(v);
return;
}
if (acb_is_one(a))
{
acb_hypgeom_pfq_direct(res, NULL, 0, b, 1, z, -1, prec);
}
else
{
acb_struct c[3];
c[0] = *a;
c[1] = *b;
acb_init(c + 2);
acb_one(c + 2);
acb_hypgeom_pfq_direct(res, c, 1, c + 1, 2, z, -1, prec);
acb_clear(c + 2);
}
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("u_asymp....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
acb_t a, b, a2, b2, z, U1, U2, t, u, M1, M2, am;
acb_struct bm[2];
ulong n1, n2;
slong prec0, prec1, prec2;
acb_init(a);
acb_init(b);
acb_init(a2);
acb_init(b2);
acb_init(z);
acb_init(U1);
acb_init(U2);
acb_init(t);
acb_init(u);
acb_init(M1);
acb_init(M2);
acb_init(am);
acb_init(bm);
acb_init(bm + 1);
if (n_randint(state, 4) == 0)
{
n1 = n_randint(state, 200);
n2 = n_randint(state, 200);
prec0 = 2 + n_randint(state, 1000);
prec1 = 2 + n_randint(state, 1000);
prec2 = 2 + n_randint(state, 1000);
}
else
{
n1 = n_randint(state, 40);
n2 = n_randint(state, 40);
prec0 = 2 + n_randint(state, 300);
prec1 = 2 + n_randint(state, 300);
prec2 = 2 + n_randint(state, 300);
}
acb_randtest_param(a, state, prec0, 1 + n_randint(state, 20));
if (n_randint(state, 4) == 0)
acb_add_ui(b, a, n_randint(state, 10), prec0);
else
acb_randtest_param(b, state, prec0, 1 + n_randint(state, 20));
acb_randtest(z, state, prec0, 1 + n_randint(state, 20));
/* Test Kummer's transformation */
acb_sub(a2, a, b, prec0);
acb_add_ui(a2, a2, 1, prec0);
acb_sub_ui(b2, b, 2, prec0);
acb_neg(b2, b2);
acb_hypgeom_u_asymp(U1, a, b, z, n1, prec1);
acb_hypgeom_u_asymp(U2, a2, b2, z, n2, prec2);
if (!acb_overlaps(U1, U2))
{
flint_printf("FAIL (Kummer transformation)\n");
flint_printf("iter = %wd\n", iter);
flint_printf("a = "); acb_printd(a, 50); flint_printf("\n");
flint_printf("b = "); acb_printd(b, 50); flint_printf("\n");
flint_printf("z = "); acb_printd(z, 50); flint_printf("\n");
flint_printf("n1 = %wd, n2 = %wd, prec1 = %wd, prec2 = %wd\n", n1, n2, prec1, prec2);
flint_printf("U1 = "); acb_printd(U1, 100); flint_printf("\n");
flint_printf("U2 = "); acb_printd(U2, 100); flint_printf("\n");
abort();
}
/* Check contiguous relation
(b-a-1)U(a,b-1,z) + z U(a,b+1,z) + (1-b-z) U(a,b,z) = 0 */
acb_one(t);
acb_sub(t, t, b, prec0);
acb_sub(t, t, z, prec0);
acb_mul(u, U1, t, prec1);
acb_add_ui(b2, b, 1, prec0);
acb_hypgeom_u_asymp(U2, a, b2, z, n2, prec2);
acb_addmul(u, U2, z, prec1);
acb_sub_ui(b2, b, 1, prec0);
acb_hypgeom_u_asymp(U2, a, b2, z, n2, prec2);
acb_sub(t, b, a, prec0);
acb_sub_ui(t, t, 1, prec0);
acb_mul(t, t, U2, prec0);
acb_add(t, t, u, prec0);
if (!acb_contains_zero(t))
{
flint_printf("FAIL (contiguous relation)\n");
flint_printf("iter = %wd\n", iter);
flint_printf("a = "); acb_printd(a, 50); flint_printf("\n");
flint_printf("b = "); acb_printd(b, 50); flint_printf("\n");
flint_printf("z = "); acb_printd(z, 50); flint_printf("\n");
flint_printf("n1 = %wd, n2 = %wd, prec1 = %wd, prec2 = %wd\n", n1, n2, prec1, prec2);
flint_printf("U1 = "); acb_printd(U1, 100); flint_printf("\n");
flint_printf("t = "); acb_printd(t, 100); flint_printf("\n");
abort();
}
/* Test U(a,b,z) = gamma(1-b)/gamma(a-b+1) M(a,b,z)
+ gamma(b-1)/gamma(a) z^(1-b) M(a-b+1,2-b,z) */
acb_set(am, a);
acb_set(bm, b);
acb_one(bm + 1);
acb_hypgeom_pfq_direct(M1, am, 1, bm, 2, z, n2, prec2);
acb_sub(am, a, b, prec2);
acb_add_ui(am, am, 1, prec2);
acb_sub_ui(bm, b, 2, prec2);
acb_neg(bm, bm);
acb_one(bm + 1);
acb_hypgeom_pfq_direct(M2, am, 1, bm, 2, z, n2, prec2);
acb_sub(am, a, b, prec2);
acb_add_ui(am, am, 1, prec2);
acb_rgamma(am, am, prec2);
acb_mul(M1, M1, am, prec2);
acb_sub_ui(am, b, 1, prec2);
acb_neg(am, am);
acb_gamma(am, am, prec2);
acb_mul(M1, M1, am, prec2);
acb_rgamma(am, a, prec2);
acb_mul(M2, M2, am, prec2);
acb_sub_ui(am, b, 1, prec2);
acb_gamma(am, am, prec2);
acb_mul(M2, M2, am, prec2);
acb_sub_ui(am, b, 1, prec2);
acb_neg(am, am);
acb_pow(am, z, am, prec2);
acb_mul(M2, M2, am, prec2);
acb_add(U2, M1, M2, prec2);
acb_pow(am, z, a, prec2);
acb_mul(U2, U2, am, prec2);
if (!acb_overlaps(U1, U2))
{
flint_printf("FAIL (U in terms of M)\n");
flint_printf("iter = %wd\n", iter);
flint_printf("a = "); acb_printd(a, 50); flint_printf("\n");
flint_printf("b = "); acb_printd(b, 50); flint_printf("\n");
flint_printf("z = "); acb_printd(z, 50); flint_printf("\n");
flint_printf("n1 = %wd, n2 = %wd, prec1 = %wd, prec2 = %wd\n", n1, n2, prec1, prec2);
flint_printf("U1 = "); acb_printd(U1, 100); flint_printf("\n");
flint_printf("U2 = "); acb_printd(U2, 100); flint_printf("\n");
abort();
}
/* Test special value: b = a+1 */
acb_add_ui(b, a, 1, prec0);
acb_hypgeom_u_asymp(U1, a, b, z, n1, prec1);
acb_one(U2);
if (!acb_overlaps(U1, U2))
{
flint_printf("FAIL (special value)\n");
flint_printf("iter = %wd\n", iter);
flint_printf("a = "); acb_printd(a, 50); flint_printf("\n");
flint_printf("b = "); acb_printd(b, 50); flint_printf("\n");
flint_printf("z = "); acb_printd(z, 50); flint_printf("\n");
flint_printf("n1 = %wd, n2 = %wd, prec1 = %wd, prec2 = %wd\n", n1, n2, prec1, prec2);
flint_printf("U1 = "); acb_printd(U1, 100); flint_printf("\n");
flint_printf("U2 = "); acb_printd(U2, 100); flint_printf("\n");
abort();
}
acb_clear(a);
acb_clear(b);
acb_clear(a2);
acb_clear(b2);
acb_clear(z);
acb_clear(U1);
acb_clear(U2);
acb_clear(t);
acb_clear(u);
acb_clear(M1);
acb_clear(M2);
acb_clear(am);
acb_clear(bm);
acb_clear(bm + 1);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
acb_hypgeom_u_1f1(acb_t res, const acb_t a, const acb_t b, const acb_t z, long prec)
{
if (acb_is_int(b))
{
acb_poly_t aa, bb, zz;
acb_poly_init(aa);
acb_poly_init(bb);
acb_poly_init(zz);
acb_poly_set_acb(aa, a);
acb_poly_set_coeff_acb(bb, 0, b);
acb_poly_set_coeff_si(bb, 1, 1);
acb_poly_set_acb(zz, z);
acb_hypgeom_u_1f1_series(zz, aa, bb, zz, 1, prec);
acb_poly_get_coeff_acb(res, zz, 0);
acb_poly_clear(aa);
acb_poly_clear(bb);
acb_poly_clear(zz);
}
else
{
acb_t t, u, v;
acb_struct aa[3];
acb_init(t);
acb_init(u);
acb_init(v);
acb_init(aa + 0);
acb_init(aa + 1);
acb_init(aa + 2);
acb_set(aa, a);
acb_set(aa + 1, b);
acb_one(aa + 2);
acb_hypgeom_pfq_direct(u, aa, 1, aa + 1, 2, z, -1, prec);
acb_sub(aa, a, b, prec);
acb_add_ui(aa, aa, 1, prec);
acb_sub_ui(aa + 1, b, 2, prec);
acb_neg(aa + 1, aa + 1);
acb_hypgeom_pfq_direct(v, aa, 1, aa + 1, 2, z, -1, prec);
acb_sub_ui(aa + 1, b, 1, prec);
/* rgamma(a-b+1) * gamma(1-b) * u */
acb_rgamma(t, aa, prec);
acb_mul(u, u, t, prec);
acb_neg(t, aa + 1);
acb_gamma(t, t, prec);
acb_mul(u, u, t, prec);
/* rgamma(a) * gamma(b-1) * z^(1-b) * v */
acb_rgamma(t, a, prec);
acb_mul(v, v, t, prec);
acb_gamma(t, aa + 1, prec);
acb_mul(v, v, t, prec);
acb_neg(t, aa + 1);
acb_pow(t, z, t, prec);
acb_mul(v, v, t, prec);
acb_add(res, u, v, prec);
acb_clear(t);
acb_clear(u);
acb_clear(v);
acb_clear(aa + 0);
acb_clear(aa + 1);
acb_clear(aa + 2);
}
}