1 /*
2 Copyright (C) 2015 Fredrik Johansson
3
4 This file is part of Arb.
5
6 Arb is free software: you can redistribute it and/or modify it under
7 the terms of the GNU Lesser General Public License (LGPL) as published
8 by the Free Software Foundation; either version 2.1 of the License, or
9 (at your option) any later version. See <http://www.gnu.org/licenses/>.
10 */
11
12 #include "acb.h"
13 #include "acb_poly.h"
14
15 void
_arb_const_zeta_minus_one_eval(arb_t y,slong prec)16 _arb_const_zeta_minus_one_eval(arb_t y, slong prec)
17 {
18 acb_struct z[2];
19 acb_t s, a;
20 acb_init(z + 0);
21 acb_init(z + 1);
22 acb_init(s);
23 acb_init(a);
24 acb_set_si(s, -1);
25 acb_one(a);
26 _acb_poly_zeta_cpx_series(z, s, a, 0, 2, prec + 20);
27 arb_set(y, acb_realref(z + 1));
28 acb_clear(z + 0);
29 acb_clear(z + 1);
30 acb_clear(s);
31 acb_clear(a);
32 }
33
ARB_DEF_CACHED_CONSTANT(_arb_const_zeta_minus_one,_arb_const_zeta_minus_one_eval)34 ARB_DEF_CACHED_CONSTANT(_arb_const_zeta_minus_one, _arb_const_zeta_minus_one_eval)
35
36 /* LogG(z) = (z-1) LogGamma(z) - zeta'(-1,z) + zeta'(-1)
37 LogG'(z) = (1/2)(-2z + 1 + log(2pi)) + (z-1) digamma(z) */
38
39 void
40 _acb_log_barnes_g_zeta(acb_t res, const acb_t z, slong prec)
41 {
42 acb_struct t[3];
43 acb_init(t + 0);
44 acb_init(t + 1);
45 acb_init(t + 2);
46
47 acb_set_si(t + 2, -1);
48 _acb_poly_zeta_cpx_series(t, t + 2, z, 0, 2, prec);
49
50 _arb_const_zeta_minus_one(acb_realref(t), prec);
51 arb_zero(acb_imagref(t));
52 acb_sub(t, t, t + 1, prec);
53
54 acb_lgamma(t + 1, z, prec);
55 acb_sub_ui(t + 2, z, 1, prec);
56 acb_addmul(t, t + 1, t + 2, prec);
57
58 acb_set(res, t);
59
60 acb_clear(t + 0);
61 acb_clear(t + 1);
62 acb_clear(t + 2);
63 }
64
65 void
_acb_barnes_g_ui_rec(acb_t res,ulong n,slong prec)66 _acb_barnes_g_ui_rec(acb_t res, ulong n, slong prec)
67 {
68 acb_t t;
69 ulong k;
70
71 acb_init(t);
72
73 acb_one(res);
74 acb_one(t);
75
76 for (k = 3; k < n; k++)
77 {
78 acb_mul_ui(t, t, k - 1, prec);
79 acb_mul(res, res, t, prec);
80 }
81
82 acb_clear(t);
83 }
84
85 void
acb_log_barnes_g(acb_t res,const acb_t z,slong prec)86 acb_log_barnes_g(acb_t res, const acb_t z, slong prec)
87 {
88 if (acb_is_int(z))
89 {
90 if (arb_is_nonpositive(acb_realref(z)))
91 {
92 acb_indeterminate(res);
93 return;
94 }
95
96 if (arf_cmpabs_ui(arb_midref(acb_realref(z)), prec) < 0)
97 {
98 _acb_barnes_g_ui_rec(res,
99 arf_get_si(arb_midref(acb_realref(z)), ARF_RND_DOWN), prec);
100 acb_log(res, res, prec);
101 return;
102 }
103 }
104
105 _acb_log_barnes_g_zeta(res, z, prec);
106 }
107
108 void
acb_barnes_g(acb_t res,const acb_t z,slong prec)109 acb_barnes_g(acb_t res, const acb_t z, slong prec)
110 {
111 int real;
112
113 real = acb_is_real(z);
114
115 if (acb_is_int(z))
116 {
117 if (arb_is_nonpositive(acb_realref(z)))
118 {
119 acb_zero(res);
120 return;
121 }
122
123 if (arf_cmpabs_ui(arb_midref(acb_realref(z)), prec) < 0)
124 {
125 _acb_barnes_g_ui_rec(res,
126 arf_get_si(arb_midref(acb_realref(z)), ARF_RND_DOWN), prec);
127 return;
128 }
129 }
130
131 _acb_log_barnes_g_zeta(res, z, prec);
132 acb_exp(res, res, prec);
133
134 if (real)
135 arb_zero(acb_imagref(res));
136 }
137
138