void
_acb_hypgeom_coulomb_series(acb_ptr F, acb_ptr G,
acb_ptr Hpos, acb_ptr Hneg, const acb_t l, const acb_t eta,
acb_srcptr z, slong zlen, slong len, slong prec)
{
acb_ptr t, v;
if (len == 0)
return;
zlen = FLINT_MIN(zlen, len);
if (zlen == 1)
{
acb_hypgeom_coulomb(F, G, Hpos, Hneg, l, eta, z, prec);
if (F != NULL) _acb_vec_zero(F + 1, len - 1);
if (G != NULL) _acb_vec_zero(G + 1, len - 1);
if (Hpos != NULL) _acb_vec_zero(Hpos + 1, len - 1);
if (Hneg != NULL) _acb_vec_zero(Hneg + 1, len - 1);
return;
}
t = _acb_vec_init(len);
v = _acb_vec_init(zlen);
/* copy nonconstant part first to allow aliasing */
acb_zero(v);
_acb_vec_set(v + 1, z + 1, zlen - 1);
acb_hypgeom_coulomb_jet(F, G, Hpos, Hneg, l, eta, z, len, prec);
if (F != NULL)
{
_acb_vec_set(t, F, len);
_acb_poly_compose_series(F, t, len, v, zlen, len, prec);
}
if (G != NULL)
{
_acb_vec_set(t, G, len);
_acb_poly_compose_series(G, t, len, v, zlen, len, prec);
}
if (Hpos != NULL)
{
_acb_vec_set(t, Hpos, len);
_acb_poly_compose_series(Hpos, t, len, v, zlen, len, prec);
}
if (Hneg != NULL)
{
_acb_vec_set(t, Hneg, len);
_acb_poly_compose_series(Hneg, t, len, v, zlen, len, prec);
}
_acb_vec_clear(t, len);
_acb_vec_clear(v, zlen);
}
void
acb_poly_compose_series(acb_poly_t res,
const acb_poly_t poly1,
const acb_poly_t poly2, slong n, slong prec)
{
slong len1 = poly1->length;
slong len2 = poly2->length;
slong lenr;
if (len2 != 0 && !acb_is_zero(poly2->coeffs))
{
flint_printf("exception: compose_series: inner "
"polynomial must have zero constant term\n");
abort();
}
if (len1 == 0 || n == 0)
{
acb_poly_zero(res);
return;
}
if (len2 == 0 || len1 == 1)
{
acb_poly_set_acb(res, poly1->coeffs);
return;
}
lenr = FLINT_MIN((len1 - 1) * (len2 - 1) + 1, n);
len1 = FLINT_MIN(len1, lenr);
len2 = FLINT_MIN(len2, lenr);
if ((res != poly1) && (res != poly2))
{
acb_poly_fit_length(res, lenr);
_acb_poly_compose_series(res->coeffs, poly1->coeffs, len1,
poly2->coeffs, len2, lenr, prec);
_acb_poly_set_length(res, lenr);
_acb_poly_normalise(res);
}
else
{
acb_poly_t t;
acb_poly_init2(t, lenr);
_acb_poly_compose_series(t->coeffs, poly1->coeffs, len1,
poly2->coeffs, len2, lenr, prec);
_acb_poly_set_length(t, lenr);
_acb_poly_normalise(t);
acb_poly_swap(res, t);
acb_poly_clear(t);
}
}
void
_acb_dirichlet_hardy_z_series(acb_ptr res, acb_srcptr s, slong slen,
const dirichlet_group_t G, const dirichlet_char_t chi,
slong len, slong prec)
{
slen = FLINT_MIN(slen, len);
if (len == 0)
return;
if (slen == 1)
{
acb_dirichlet_hardy_z(res, s, G, chi, 1, prec);
_acb_vec_zero(res + 1, len - 1);
}
else
{
acb_ptr t, u;
t = _acb_vec_init(len);
u = _acb_vec_init(slen);
acb_dirichlet_hardy_z(t, s, G, chi, len, prec);
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, s + 1, slen - 1);
_acb_poly_compose_series(res, t, len, u, slen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, slen);
}
}
void
_acb_poly_zeta_series(acb_ptr res, acb_srcptr h, slong hlen, const acb_t a, int deflate, slong len, slong prec)
{
acb_ptr t, u;
hlen = FLINT_MIN(hlen, len);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
_acb_poly_zeta_cpx_reflect(t, h, a, deflate, len, prec);
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, t, len, u, hlen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
void
_acb_poly_agm1_series(acb_ptr res, acb_srcptr z, long zlen, long len, long prec)
{
acb_ptr t, u;
zlen = FLINT_MIN(zlen, len);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
acb_agm1_cpx(t, z, len, prec);
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, z + 1, zlen - 1);
_acb_poly_compose_series(res, t, len, u, zlen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
void
_acb_poly_rgamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong len, slong prec)
{
int reflect;
slong i, rflen, r, n, wp;
acb_ptr t, u, v;
acb_struct f[2];
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
acb_rgamma(res, h, prec);
_acb_vec_zero(res + 1, len - 1);
return;
}
/* use real code for real input */
if (_acb_vec_is_real(h, hlen))
{
arb_ptr tmp = _arb_vec_init(len);
for (i = 0; i < hlen; i++)
arb_set(tmp + i, acb_realref(h + i));
_arb_poly_rgamma_series(tmp, tmp, hlen, len, prec);
for (i = 0; i < len; i++)
acb_set_arb(res + i, tmp + i);
_arb_vec_clear(tmp, len);
return;
}
wp = prec + FLINT_BIT_COUNT(prec);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
v = _acb_vec_init(len);
acb_init(f);
acb_init(f + 1);
/* otherwise use Stirling series */
acb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp);
/* rgamma(h) = (gamma(1-h+r) sin(pi h)) / (rf(1-h, r) * pi), h = h0 + t*/
if (reflect)
{
/* u = gamma(r+1-h) */
acb_sub_ui(f, h, r + 1, wp);
acb_neg(f, f);
_acb_poly_gamma_stirling_eval(t, f, n, len, wp);
_acb_poly_exp_series(u, t, len, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(u + i, u + i);
/* v = sin(pi x) */
acb_set(f, h);
acb_one(f + 1);
_acb_poly_sin_pi_series(v, f, 2, len, wp);
_acb_poly_mullow(t, u, len, v, len, len, wp);
/* rf(1-h,r) * pi */
if (r == 0)
{
acb_const_pi(u, wp);
_acb_vec_scalar_div(v, t, len, u, wp);
}
else
{
acb_sub_ui(f, h, 1, wp);
acb_neg(f, f);
acb_set_si(f + 1, -1);
rflen = FLINT_MIN(len, r + 1);
_acb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r, rflen, wp);
acb_const_pi(u, wp);
_acb_vec_scalar_mul(v, v, rflen, u, wp);
/* divide by rising factorial */
/* TODO: might better to use div_series, when it has a good basecase */
_acb_poly_inv_series(u, v, rflen, len, wp);
_acb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* rgamma(h) = rgamma(h+r) rf(h,r) */
if (r == 0)
{
acb_add_ui(f, h, r, wp);
_acb_poly_gamma_stirling_eval(t, f, n, len, wp);
_acb_vec_neg(t, t, len);
_acb_poly_exp_series(v, t, len, len, wp);
}
else
{
acb_set(f, h);
acb_one(f + 1);
rflen = FLINT_MIN(len, r + 1);
_acb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, wp);
acb_add_ui(f, h, r, wp);
_acb_poly_gamma_stirling_eval(v, f, n, len, wp);
_acb_vec_neg(v, v, len);
_acb_poly_exp_series(u, v, len, len, wp);
_acb_poly_mullow(v, u, len, t, rflen, len, wp);
}
}
/* compose with nonconstant part */
acb_zero(t);
_acb_vec_set(t + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, v, len, t, hlen, len, prec);
acb_clear(f);
acb_clear(f + 1);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
_acb_vec_clear(v, len);
}
void
_acb_poly_zeta_series(acb_ptr res, acb_srcptr h, long hlen, const acb_t a, int deflate, long len, long prec)
{
long i;
acb_ptr t, u;
hlen = FLINT_MIN(hlen, len);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
/* use reflection formula */
if (arf_sgn(arb_midref(acb_realref(h))) < 0 && acb_is_one(a))
{
/* zeta(s) = (2*pi)**s * sin(pi*s/2) / pi * gamma(1-s) * zeta(1-s) */
acb_t pi;
acb_ptr f, s1, s2, s3, s4;
acb_init(pi);
f = _acb_vec_init(2);
s1 = _acb_vec_init(len);
s2 = _acb_vec_init(len);
s3 = _acb_vec_init(len);
s4 = _acb_vec_init(len);
acb_const_pi(pi, prec);
/* s1 = (2*pi)**s */
acb_mul_2exp_si(pi, pi, 1);
_acb_poly_pow_cpx(s1, pi, h, len, prec);
acb_mul_2exp_si(pi, pi, -1);
/* s2 = sin(pi*s/2) / pi */
acb_set(f, h);
acb_one(f + 1);
acb_mul_2exp_si(f, f, -1);
acb_mul_2exp_si(f + 1, f + 1, -1);
_acb_poly_sin_pi_series(s2, f, 2, len, prec);
_acb_vec_scalar_div(s2, s2, len, pi, prec);
/* s3 = gamma(1-s) */
acb_sub_ui(f, h, 1, prec);
acb_neg(f, f);
acb_set_si(f + 1, -1);
_acb_poly_gamma_series(s3, f, 2, len, prec);
/* s4 = zeta(1-s) */
acb_sub_ui(f, h, 1, prec);
acb_neg(f, f);
_acb_poly_zeta_cpx_series(s4, f, a, 0, len, prec);
for (i = 1; i < len; i += 2)
acb_neg(s4 + i, s4 + i);
_acb_poly_mullow(u, s1, len, s2, len, len, prec);
_acb_poly_mullow(s1, s3, len, s4, len, len, prec);
_acb_poly_mullow(t, u, len, s1, len, len, prec);
/* add 1/(1-(s+t)) = 1/(1-s) + t/(1-s)^2 + ... */
if (deflate)
{
acb_sub_ui(u, h, 1, prec);
acb_neg(u, u);
acb_inv(u, u, prec);
for (i = 1; i < len; i++)
acb_mul(u + i, u + i - 1, u, prec);
_acb_vec_add(t, t, u, len, prec);
}
acb_clear(pi);
_acb_vec_clear(f, 2);
_acb_vec_clear(s1, len);
_acb_vec_clear(s2, len);
_acb_vec_clear(s3, len);
_acb_vec_clear(s4, len);
}
else
{
_acb_poly_zeta_cpx_series(t, h, a, deflate, len, prec);
}
/* compose with nonconstant part */
acb_zero(u);
_acb_vec_set(u + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, t, len, u, hlen, len, prec);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
void
_acb_poly_lgamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong len, slong prec)
{
int reflect;
slong i, r, n, wp;
acb_t zr;
acb_ptr t, u;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
acb_lgamma(res, h, prec);
if (acb_is_finite(res))
_acb_vec_zero(res + 1, len - 1);
else
_acb_vec_indeterminate(res + 1, len - 1);
return;
}
if (len == 2)
{
acb_t v;
acb_init(v);
acb_set(v, h + 1);
acb_digamma(res + 1, h, prec);
acb_lgamma(res, h, prec);
acb_mul(res + 1, res + 1, v, prec);
acb_clear(v);
return;
}
/* use real code for real input and output */
if (_acb_vec_is_real(h, hlen) && arb_is_positive(acb_realref(h)))
{
arb_ptr tmp = _arb_vec_init(len);
for (i = 0; i < hlen; i++)
arb_set(tmp + i, acb_realref(h + i));
_arb_poly_lgamma_series(tmp, tmp, hlen, len, prec);
for (i = 0; i < len; i++)
acb_set_arb(res + i, tmp + i);
_arb_vec_clear(tmp, len);
return;
}
wp = prec + FLINT_BIT_COUNT(prec);
t = _acb_vec_init(len);
u = _acb_vec_init(len);
acb_init(zr);
/* use Stirling series */
acb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp);
if (reflect)
{
/* log gamma(h+x) = log rf(1-(h+x), r) - log gamma(1-(h+x)+r) - log sin(pi (h+x)) + log(pi) */
if (r != 0) /* otherwise t = 0 */
{
acb_sub_ui(u, h, 1, wp);
acb_neg(u, u);
_log_rising_ui_series(t, u, r, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(t + i, t + i);
}
acb_sub_ui(u, h, 1, wp);
acb_neg(u, u);
acb_add_ui(zr, u, r, wp);
_acb_poly_gamma_stirling_eval(u, zr, n, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(u + i, u + i);
_acb_vec_sub(t, t, u, len, wp);
/* log(sin) is unstable with large imaginary parts;
cot_pi is implemented in a numerically stable way */
acb_set(u, h);
acb_one(u + 1);
_acb_poly_cot_pi_series(u, u, 2, len - 1, wp);
_acb_poly_integral(u, u, len, wp);
acb_const_pi(u, wp);
_acb_vec_scalar_mul(u + 1, u + 1, len - 1, u, wp);
acb_log_sin_pi(u, h, wp);
_acb_vec_sub(u, t, u, len, wp);
acb_const_pi(t, wp); /* todo: constant for log pi */
acb_log(t, t, wp);
acb_add(u, u, t, wp);
}
else
{
/* log gamma(x) = log gamma(x+r) - log rf(x,r) */
acb_add_ui(zr, h, r, wp);
_acb_poly_gamma_stirling_eval(u, zr, n, len, wp);
if (r != 0)
{
_log_rising_ui_series(t, h, r, len, wp);
_acb_vec_sub(u, u, t, len, wp);
}
}
/* compose with nonconstant part */
acb_zero(t);
_acb_vec_set(t + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, u, len, t, hlen, len, prec);
acb_clear(zr);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
}
void
_acb_poly_digamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong len, slong prec)
{
int reflect;
slong i, r, n, rflen, wp;
acb_t zr;
acb_ptr t, u, v;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
acb_digamma(res, h, prec);
if (acb_is_finite(res))
_acb_vec_zero(res + 1, len - 1);
else
_acb_vec_indeterminate(res + 1, len - 1);
return;
}
/* use real code for real input */
if (_acb_vec_is_real(h, hlen))
{
arb_ptr tmp = _arb_vec_init(len);
for (i = 0; i < hlen; i++)
arb_set(tmp + i, acb_realref(h + i));
_arb_poly_digamma_series(tmp, tmp, hlen, len, prec);
for (i = 0; i < len; i++)
acb_set_arb(res + i, tmp + i);
_arb_vec_clear(tmp, len);
return;
}
wp = prec + FLINT_BIT_COUNT(prec);
t = _acb_vec_init(len + 1);
u = _acb_vec_init(len + 1);
v = _acb_vec_init(len + 1);
acb_init(zr);
/* use Stirling series */
acb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 1, wp);
/* psi(x) = psi((1-x)+r) - h(1-x,r) - pi*cot(pi*x) */
if (reflect)
{
if (r != 0) /* otherwise t = 0 */
{
acb_sub_ui(v, h, 1, wp);
acb_neg(v, v);
acb_one(v + 1);
rflen = FLINT_MIN(len + 1, r + 1);
_acb_poly_rising_ui_series(u, v, 2, r, rflen, wp);
_acb_poly_derivative(v, u, rflen, wp);
_acb_poly_div_series(t, v, rflen - 1, u, rflen, len, wp);
for (i = 1; i < len; i += 2)
acb_neg(t + i, t + i);
}
acb_sub_ui(zr, h, r + 1, wp);
acb_neg(zr, zr);
_acb_poly_gamma_stirling_eval2(u, zr, n, len + 1, 1, wp);
for (i = 1; i < len; i += 2)
acb_neg(u + i, u + i);
_acb_vec_sub(u, u, t, len, wp);
acb_set(t, h);
acb_one(t + 1);
_acb_poly_cot_pi_series(t, t, 2, len, wp);
acb_const_pi(v, wp);
_acb_vec_scalar_mul(t, t, len, v, wp);
_acb_vec_sub(u, u, t, len, wp);
}
else
{
if (r == 0)
{
acb_add_ui(zr, h, r, wp);
_acb_poly_gamma_stirling_eval2(u, zr, n, len + 1, 1, wp);
}
else
{
acb_set(v, h);
acb_one(v + 1);
rflen = FLINT_MIN(len + 1, r + 1);
_acb_poly_rising_ui_series(u, v, 2, r, rflen, wp);
_acb_poly_derivative(v, u, rflen, wp);
_acb_poly_div_series(t, v, rflen - 1, u, rflen, len, wp);
acb_add_ui(zr, h, r, wp);
_acb_poly_gamma_stirling_eval2(u, zr, n, len + 1, 1, wp);
_acb_vec_sub(u, u, t, len, wp);
}
}
/* compose with nonconstant part */
acb_zero(t);
_acb_vec_set(t + 1, h + 1, hlen - 1);
_acb_poly_compose_series(res, u, len, t, hlen, len, prec);
acb_clear(zr);
_acb_vec_clear(t, len + 1);
_acb_vec_clear(u, len + 1);
_acb_vec_clear(v, len + 1);
}