static __inline__ void _log_rising_ui_series(acb_ptr t, const acb_t x, slong r, slong len, slong prec) { acb_struct f[2]; slong rflen; acb_init(f); acb_init(f + 1); acb_set(f, x); acb_one(f + 1); rflen = FLINT_MIN(len, r + 1); _acb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, prec); _acb_poly_log_series(t, t, rflen, len, prec); _acb_log_rising_correct_branch(t, t, x, r, prec); acb_clear(f); acb_clear(f + 1); }
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_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); }