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implement scaled modified Bessel functions

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This commit is contained in:
fredrik 2018-03-23 13:41:46 +01:00
parent 7a4a7e10e4
commit 7f98274601
11 changed files with 250 additions and 59 deletions
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12
acb_hypgeom.h
View file

@ -111,14 +111,16 @@ void acb_hypgeom_bessel_j_0f1(acb_t res, const acb_t nu, const acb_t z, slong pr
void acb_hypgeom_bessel_j_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_j(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec);
void acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec);
void acb_hypgeom_bessel_i(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_i_scaled(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_k_0f1_series(acb_poly_t res, const acb_poly_t n, const acb_poly_t z, slong len, slong prec);
void acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec);
void acb_hypgeom_bessel_k_0f1_series(acb_poly_t res, const acb_poly_t n, const acb_poly_t z, int scaled, slong len, slong prec);
void acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec);
void acb_hypgeom_bessel_k(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_k_scaled(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_y(acb_t res, const acb_t nu, const acb_t z, slong prec);
void acb_hypgeom_bessel_jy(acb_t res1, acb_t res2, const acb_t nu, const acb_t z, slong prec);

2
acb_hypgeom/0f1.c
View file

@ -37,7 +37,7 @@ acb_hypgeom_0f1_asymp(acb_t res, const acb_t a, const acb_t z, int regularized,
if (neg)
acb_hypgeom_bessel_j_asymp(v, u, v, prec);
else
acb_hypgeom_bessel_i_asymp(v, u, v, prec);
acb_hypgeom_bessel_i_asymp(v, u, v, 0, prec);
acb_neg(u, u);
acb_pow(t, t, u, prec);

56
acb_hypgeom/bessel_i.c
View file

@ -13,7 +13,7 @@
void
acb_hypgeom_bessel_i_asymp_prefactors(acb_t A, acb_t B, acb_t C,
const acb_t nu, const acb_t z, slong prec)
const acb_t nu, const acb_t z, int scaled, slong prec)
{
acb_t t, u;
@ -53,15 +53,26 @@ acb_hypgeom_bessel_i_asymp_prefactors(acb_t A, acb_t B, acb_t C,
arb_union(acb_imagref(t), acb_imagref(t), acb_imagref(u), prec);
}
acb_exp_invexp(B, A, z, prec);
acb_mul(A, A, t, prec);
if (scaled)
{
acb_neg(u, z);
acb_mul_2exp_si(u, u, 1);
acb_exp(u, u, prec);
acb_mul(A, t, u, prec);
acb_one(B);
}
else
{
acb_exp_invexp(B, A, z, prec);
acb_mul(A, A, t, prec);
}
acb_clear(t);
acb_clear(u);
}
void
acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
{
acb_t A1, A2, C, U1, U2, s, t, u;
int is_real, is_imag;
@ -89,7 +100,10 @@ acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
is_imag = 1;
}
acb_hypgeom_bessel_i_asymp_prefactors(A1, A2, C, nu, z, prec);
if (scaled)
is_imag = 0;
acb_hypgeom_bessel_i_asymp_prefactors(A1, A2, C, nu, z, scaled, prec);
/* todo: if Ap ~ 2^a and Am = 2^b and U1 ~ U2 ~ 1, change precision? */
@ -134,7 +148,7 @@ acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
}
void
acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
{
acb_struct b[2];
acb_t w, c, t;
@ -143,7 +157,7 @@ acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
{
acb_init(t);
acb_neg(t, nu);
acb_hypgeom_bessel_i_0f1(res, t, z, prec);
acb_hypgeom_bessel_i_0f1(res, t, z, scaled, prec);
acb_clear(t);
return;
}
@ -169,6 +183,13 @@ acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
acb_hypgeom_pfq_direct(t, NULL, 0, b, 2, w, -1, prec);
if (scaled)
{
acb_neg(w, z);
acb_exp(w, w, prec);
acb_mul(t, t, w, prec);
}
acb_mul(res, t, c, prec);
acb_clear(b + 0);
@ -188,9 +209,26 @@ acb_hypgeom_bessel_i(acb_t res, const acb_t nu, const acb_t z, slong prec)
if (mag_cmp_2exp_si(zmag, 4) < 0 ||
(mag_cmp_2exp_si(zmag, 64) < 0 && 2 * mag_get_d(zmag) < prec))
acb_hypgeom_bessel_i_0f1(res, nu, z, prec);
acb_hypgeom_bessel_i_0f1(res, nu, z, 0, prec);
else
acb_hypgeom_bessel_i_asymp(res, nu, z, prec);
acb_hypgeom_bessel_i_asymp(res, nu, z, 0, prec);
mag_clear(zmag);
}
void
acb_hypgeom_bessel_i_scaled(acb_t res, const acb_t nu, const acb_t z, slong prec)
{
mag_t zmag;
mag_init(zmag);
acb_get_mag(zmag, z);
if (mag_cmp_2exp_si(zmag, 4) < 0 ||
(mag_cmp_2exp_si(zmag, 64) < 0 && 2 * mag_get_d(zmag) < prec))
acb_hypgeom_bessel_i_0f1(res, nu, z, 1, prec);
else
acb_hypgeom_bessel_i_asymp(res, nu, z, 1, prec);
mag_clear(zmag);
}

56
acb_hypgeom/bessel_k.c
View file

@ -12,7 +12,7 @@
#include "acb_hypgeom.h"
void
acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
{
acb_t t, a, b, w;
@ -32,9 +32,12 @@ acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
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);
if (!scaled)
{
acb_neg(a, z);
acb_exp(a, a, prec);
acb_mul(t, t, a, prec);
}
acb_mul_2exp_si(w, z, 1);
acb_rsqrt(w, w, prec);
@ -52,7 +55,7 @@ acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
void
acb_hypgeom_bessel_k_0f1_series(acb_poly_t res,
const acb_poly_t nu, const acb_poly_t z,
slong len, slong prec)
int scaled, slong len, slong prec)
{
acb_poly_t s, u, A, B;
acb_poly_struct b[2];
@ -101,6 +104,12 @@ acb_hypgeom_bessel_k_0f1_series(acb_poly_t res,
acb_poly_shift_right(B, B, 1);
}
if (scaled)
{
acb_poly_exp_series(u, z, len, prec);
acb_poly_mullow(A, A, u, len, prec);
}
acb_poly_div_series(res, A, B, len, prec);
arb_const_pi(c, prec);
@ -116,7 +125,7 @@ acb_hypgeom_bessel_k_0f1_series(acb_poly_t res,
}
void
acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
{
if (acb_is_int(nu))
{
@ -130,7 +139,7 @@ acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
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_hypgeom_bessel_k_0f1_series(rx, nux, zx, scaled, 1, prec);
acb_poly_get_coeff_acb(res, rx, 0);
@ -176,8 +185,16 @@ acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, slong 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_sub(v, v, u, prec);
acb_mul_2exp_si(v, v, -1);
if (scaled)
{
acb_exp(t, z, prec);
acb_mul(v, v, t, prec);
}
acb_set(res, v);
acb_clear(t);
acb_clear(u);
@ -198,9 +215,26 @@ acb_hypgeom_bessel_k(acb_t res, const acb_t nu, const acb_t z, slong prec)
if (mag_cmp_2exp_si(zmag, 4) < 0 ||
(mag_cmp_2exp_si(zmag, 64) < 0 && 2 * mag_get_d(zmag) < prec))
acb_hypgeom_bessel_k_0f1(res, nu, z, prec);
acb_hypgeom_bessel_k_0f1(res, nu, z, 0, prec);
else
acb_hypgeom_bessel_k_asymp(res, nu, z, prec);
acb_hypgeom_bessel_k_asymp(res, nu, z, 0, prec);
mag_clear(zmag);
}
void
acb_hypgeom_bessel_k_scaled(acb_t res, const acb_t nu, const acb_t z, slong prec)
{
mag_t zmag;
mag_init(zmag);
acb_get_mag(zmag, z);
if (mag_cmp_2exp_si(zmag, 4) < 0 ||
(mag_cmp_2exp_si(zmag, 64) < 0 && 2 * mag_get_d(zmag) < prec))
acb_hypgeom_bessel_k_0f1(res, nu, z, 1, prec);
else
acb_hypgeom_bessel_k_asymp(res, nu, z, 1, prec);
mag_clear(zmag);
}

54
acb_hypgeom/test/t-bessel_i.c
View file

@ -25,6 +25,7 @@ int main()
{
acb_t nu, z, jv, iv, t;
slong prec;
int scaled;
acb_init(nu);
acb_init(z);
@ -33,6 +34,7 @@ int main()
acb_init(t);
prec = 2 + n_randint(state, 500);
scaled = n_randint(state, 2);
acb_randtest_param(nu, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 10));
acb_randtest(z, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
@ -40,13 +42,16 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(iv, nu, z, prec);
acb_hypgeom_bessel_i_asymp(iv, nu, z, scaled, prec);
break;
case 1:
acb_hypgeom_bessel_i_0f1(iv, nu, z, prec);
acb_hypgeom_bessel_i_0f1(iv, nu, z, scaled, prec);
break;
default:
acb_hypgeom_bessel_i(iv, nu, z, prec);
if (scaled)
acb_hypgeom_bessel_i_scaled(iv, nu, z, prec);
else
acb_hypgeom_bessel_i(iv, nu, z, prec);
}
acb_mul_onei(t, z);
@ -57,9 +62,17 @@ int main()
acb_pow(t, t, nu, prec);
acb_div(jv, jv, t, prec);
if (scaled)
{
acb_neg(t, z);
acb_exp(t, t, prec);
acb_mul(jv, jv, t, prec);
}
if (!acb_overlaps(iv, jv))
{
flint_printf("FAIL: consistency with bessel_j\n\n");
flint_printf("scaled = %d\n\n", scaled);
flint_printf("nu = "); acb_printd(nu, 30); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("iv = "); acb_printd(iv, 30); flint_printf("\n\n");
@ -105,10 +118,10 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w0, nu0, z, prec0);
acb_hypgeom_bessel_i_asymp(w0, nu0, z, 0, prec0);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w0, nu0, z, prec0);
acb_hypgeom_bessel_i_0f1(w0, nu0, z, 0, prec0);
break;
default:
acb_hypgeom_bessel_i(w0, nu0, z, prec0);
@ -117,10 +130,10 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w1, nu0, z, prec1);
acb_hypgeom_bessel_i_asymp(w1, nu0, z, 0, prec1);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w1, nu0, z, prec1);
acb_hypgeom_bessel_i_0f1(w1, nu0, z, 0, prec1);
break;
default:
acb_hypgeom_bessel_i(w1, nu0, z, prec1);
@ -139,10 +152,10 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w1, nu1, z, prec1);
acb_hypgeom_bessel_i_asymp(w1, nu1, z, 0, prec1);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w1, nu1, z, prec1);
acb_hypgeom_bessel_i_0f1(w1, nu1, z, 0, prec1);
break;
default:
acb_hypgeom_bessel_i(w1, nu1, z, prec1);
@ -151,13 +164,22 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w2, nu2, z, prec2);
acb_hypgeom_bessel_i_asymp(w2, nu2, z, 0, prec2);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w2, nu2, z, prec2);
acb_hypgeom_bessel_i_0f1(w2, nu2, z, 0, prec2);
break;
default:
acb_hypgeom_bessel_i(w2, nu2, z, prec2);
if (n_randint(state, 2))
{
acb_hypgeom_bessel_i(w2, nu2, z, prec2);
}
else
{
acb_hypgeom_bessel_i_scaled(w2, nu2, z, prec2);
acb_exp(t, z, prec2);
acb_mul(w2, w2, t, prec2);
}
}
acb_mul(t, w1, nu1, prec0);
@ -182,10 +204,10 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w2, t, z, prec2);
acb_hypgeom_bessel_i_asymp(w2, t, z, 0, prec2);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w2, t, z, prec2);
acb_hypgeom_bessel_i_0f1(w2, t, z, 0, prec2);
break;
default:
acb_hypgeom_bessel_i(w2, t, z, prec2);
@ -197,10 +219,10 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_i_asymp(w2, t, z, prec2);
acb_hypgeom_bessel_i_asymp(w2, t, z, 0, prec2);
break;
case 1:
acb_hypgeom_bessel_i_0f1(w2, t, z, prec2);
acb_hypgeom_bessel_i_0f1(w2, t, z, 0, prec2);
break;
default:
acb_hypgeom_bessel_i(w2, t, z, prec2);

48
acb_hypgeom/test/t-bessel_k.c
View file

@ -25,6 +25,7 @@ int main()
{
acb_t nu0, nu1, nu2, z, w0, w1, w2, t, u;
slong prec0, prec1, prec2;
int scaled;
acb_init(nu0);
acb_init(nu1);
@ -40,6 +41,7 @@ int main()
prec1 = 2 + n_randint(state, 700);
prec2 = 2 + n_randint(state, 700);
scaled = n_randint(state, 2);
acb_randtest_param(nu0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_randtest(z, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_randtest(w0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
@ -55,30 +57,37 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_k_asymp(w0, nu0, z, prec0);
acb_hypgeom_bessel_k_asymp(w0, nu0, z, scaled, prec0);
break;
case 1:
acb_hypgeom_bessel_k_0f1(w0, nu0, z, prec0);
acb_hypgeom_bessel_k_0f1(w0, nu0, z, scaled, prec0);
break;
default:
acb_hypgeom_bessel_k(w0, nu0, z, prec0);
if (scaled)
acb_hypgeom_bessel_k_scaled(w0, nu0, z, prec0);
else
acb_hypgeom_bessel_k(w0, nu0, z, prec0);
}
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_k_asymp(w1, nu0, z, prec1);
acb_hypgeom_bessel_k_asymp(w1, nu0, z, scaled, prec1);
break;
case 1:
acb_hypgeom_bessel_k_0f1(w1, nu0, z, prec1);
acb_hypgeom_bessel_k_0f1(w1, nu0, z, scaled, prec1);
break;
default:
acb_hypgeom_bessel_k(w1, nu0, z, prec1);
if (scaled)
acb_hypgeom_bessel_k_scaled(w1, nu0, z, prec1);
else
acb_hypgeom_bessel_k(w1, nu0, z, prec1);
}
if (!acb_overlaps(w0, w1))
{
flint_printf("FAIL: consistency\n\n");
flint_printf("scaled = %d\n\n", scaled);
flint_printf("nu = "); acb_printd(nu0, 30); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("w0 = "); acb_printd(w0, 30); flint_printf("\n\n");
@ -89,25 +98,40 @@ int main()
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_k_asymp(w1, nu1, z, prec1);
acb_hypgeom_bessel_k_asymp(w1, nu1, z, scaled, prec1);
break;
case 1:
acb_hypgeom_bessel_k_0f1(w1, nu1, z, prec1);
acb_hypgeom_bessel_k_0f1(w1, nu1, z, scaled, prec1);
break;
default:
acb_hypgeom_bessel_k(w1, nu1, z, prec1);
if (scaled)
acb_hypgeom_bessel_k_scaled(w1, nu1, z, prec1);
else
acb_hypgeom_bessel_k(w1, nu1, z, prec1);
}
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_bessel_k_asymp(w2, nu2, z, prec2);
acb_hypgeom_bessel_k_asymp(w2, nu2, z, scaled, prec2);
break;
case 1:
acb_hypgeom_bessel_k_0f1(w2, nu2, z, prec2);
acb_hypgeom_bessel_k_0f1(w2, nu2, z, scaled, prec2);
break;
default:
acb_hypgeom_bessel_k(w2, nu2, z, prec2);
if (scaled)
{
acb_hypgeom_bessel_k(w2, nu2, z, prec2);
acb_exp(t, z, prec2);
acb_mul(w2, w2, t, prec2);
}
else
{
acb_hypgeom_bessel_k_scaled(w2, nu2, z, prec2);
acb_neg(t, z);
acb_exp(t, t, prec2);
acb_mul(w2, w2, t, prec2);
}
}
acb_mul(t, w1, nu1, prec0);

3
arb_hypgeom.h
View file

@ -74,6 +74,9 @@ void arb_hypgeom_bessel_jy(arb_t res1, arb_t res2, const arb_t nu, const arb_t z
void arb_hypgeom_bessel_i(arb_t res, const arb_t nu, const arb_t z, slong prec);
void arb_hypgeom_bessel_k(arb_t res, const arb_t nu, const arb_t z, slong prec);
void arb_hypgeom_bessel_i_scaled(arb_t res, const arb_t nu, const arb_t z, slong prec);
void arb_hypgeom_bessel_k_scaled(arb_t res, const arb_t nu, const arb_t z, slong prec);
void arb_hypgeom_airy(arb_t ai, arb_t aip, arb_t bi, arb_t bip, const arb_t z, slong prec);
void arb_hypgeom_airy_jet(arb_ptr ai, arb_ptr bi, const arb_t z, slong len, slong prec);
void arb_hypgeom_airy_series(arb_poly_t ai, arb_poly_t ai_prime,

8
arb_hypgeom/test/t-wrappers.c
View file

@ -194,10 +194,18 @@ int main()
arb_set_str(v, "[0.75761498638991 +/- 4.63e-15]", prec);
TEST(r, v, "bessel_i");
arb_hypgeom_bessel_i_scaled(r, a, z, prec);
arb_set_str(v, "[0.357871979425934 +/- 8.04e-16]", prec);
TEST(r, v, "bessel_i");
arb_hypgeom_bessel_k(r, a, z, prec);
arb_set_str(v, "[0.68361006034952 +/- 7.89e-15]", prec);
TEST(r, v, "bessel_k");
arb_hypgeom_bessel_k_scaled(r, a, z, prec);
arb_set_str(v, "[1.4472025091165 +/- 4.23e-14]", prec);
TEST(r, v, "bessel_k");
arb_hypgeom_airy(r, NULL, NULL, NULL, z, prec);
arb_set_str(v, "[0.179336305478645 +/- 2.36e-16]", prec);
TEST(r, v, "airy ai");

34
arb_hypgeom/wrappers.c
View file

@ -373,6 +373,23 @@ arb_hypgeom_bessel_i(arb_t res, const arb_t nu, const arb_t z, slong prec)
acb_clear(u);
}
void
arb_hypgeom_bessel_i_scaled(arb_t res, const arb_t nu, const arb_t z, slong prec)
{
acb_t t, u;
acb_init(t);
acb_init(u);
arb_set(acb_realref(t), nu);
arb_set(acb_realref(u), z);
acb_hypgeom_bessel_i_scaled(t, t, u, prec);
if (acb_is_finite(t) && acb_is_real(t))
arb_swap(res, acb_realref(t));
else
arb_indeterminate(res);
acb_clear(t);
acb_clear(u);
}
void
arb_hypgeom_bessel_k(arb_t res, const arb_t nu, const arb_t z, slong prec)
{
@ -390,6 +407,23 @@ arb_hypgeom_bessel_k(arb_t res, const arb_t nu, const arb_t z, slong prec)
acb_clear(u);
}
void
arb_hypgeom_bessel_k_scaled(arb_t res, const arb_t nu, const arb_t z, slong prec)
{
acb_t t, u;
acb_init(t);
acb_init(u);
arb_set(acb_realref(t), nu);
arb_set(acb_realref(u), z);
acb_hypgeom_bessel_k_scaled(t, t, u, prec);
if (acb_is_finite(t) && acb_is_real(t))
arb_swap(res, acb_realref(t));
else
arb_indeterminate(res);
acb_clear(t);
acb_clear(u);
}
void
arb_hypgeom_expint(arb_t res, const arb_t s, const arb_t z, slong prec)
{

28
doc/source/acb_hypgeom.rst
View file

@ -413,12 +413,17 @@ Bessel functions
construct the Bessel functions of the third kind (Hankel functions)
`H_{\nu}^{(1)}(z), H_{\nu}^{(2)}(z) = J_{\nu}(z) \pm i Y_{\nu}(z)`.
.. function:: void acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
Modified Bessel functions
-------------------------------------------------------------------------------
.. function:: void acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
.. function:: void acb_hypgeom_bessel_i_asymp(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
.. function:: void acb_hypgeom_bessel_i_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
.. function:: void acb_hypgeom_bessel_i(acb_t res, const acb_t nu, const acb_t z, slong prec)
.. function:: void acb_hypgeom_bessel_i_scaled(acb_t res, const acb_t nu, const acb_t z, slong prec)
Computes the modified Bessel function of the first kind
`I_{\nu}(z) = z^{\nu} (iz)^{-\nu} J_{\nu}(iz)` respectively using
asymptotic series (see :func:`acb_hypgeom_bessel_j_asymp`),
@ -431,7 +436,10 @@ Bessel functions
or an automatic algorithm choice.
.. function:: void acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
The *scaled* version computes the function `e^{-z} I_{\nu}(z)`. The *asymp*
and *0f1* functions implement both variants and allow choosing with a flag.
.. function:: void acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
Computes the modified Bessel function of the second kind via
via :func:`acb_hypgeom_u_asymp`. For all `\nu` and all `z \ne 0`, we have
@ -441,7 +449,9 @@ Bessel functions
K_{\nu}(z) = \left(\frac{2z}{\pi}\right)^{-1/2} e^{-z}
U^{*}(\nu+\tfrac{1}{2}, 2\nu+1, 2z).
.. function:: void acb_hypgeom_bessel_k_0f1_series(acb_poly_t res, const acb_poly_t nu, const acb_poly_t z, slong len, slong prec)
If *scaled* is set, computes the function `e^{z} K_{\nu}(z)`.
.. function:: void acb_hypgeom_bessel_k_0f1_series(acb_poly_t res, const acb_poly_t nu, const acb_poly_t z, int scaled, slong len, slong prec)
Computes the modified Bessel function of the second kind `K_{\nu}(z)`
as a power series truncated to length *len*,
@ -462,7 +472,9 @@ Bessel functions
As currently implemented, the output is indeterminate if `\nu[0]` is nonexact
and contains an integer.
.. function:: void acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec)
If *scaled* is set, computes the function `e^{z} K_{\nu}(z)`.
.. function:: void acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, int scaled, slong prec)
Computes the modified Bessel function of the second kind from
@ -483,11 +495,17 @@ Bessel functions
As currently implemented, the output is indeterminate if `\nu` is nonexact
and contains an integer.
If *scaled* is set, computes the function `e^{z} K_{\nu}(z)`.
.. function:: void acb_hypgeom_bessel_k(acb_t res, const acb_t nu, const acb_t z, slong prec)
Computes the modified Bessel function of the second kind `K_{\nu}(z)` using
an automatic algorithm choice.
.. function:: void acb_hypgeom_bessel_k_scaled(acb_t res, const acb_t nu, const acb_t z, slong prec)
Computes the function `e^{z} K_{\nu}(z)`.
Airy functions
-------------------------------------------------------------------------------

8
doc/source/arb_hypgeom.rst
View file

@ -278,10 +278,18 @@ Bessel functions
Computes the modified Bessel function of the first kind
`I_{\nu}(z) = z^{\nu} (iz)^{-\nu} J_{\nu}(iz)`.
.. function:: void arb_hypgeom_bessel_i_scaled(arb_t res, const arb_t nu, const arb_t z, slong prec)
Computes the function `e^{-z} I_{\nu}(z)`.
.. function:: void arb_hypgeom_bessel_k(arb_t res, const arb_t nu, const arb_t z, slong prec)
Computes the modified Bessel function of the second kind `K_{\nu}(z)`.
.. function:: void arb_hypgeom_bessel_k_scaled(arb_t res, const arb_t nu, const arb_t z, slong prec)
Computes the function `e^{z} K_{\nu}(z)`.
Airy functions
-------------------------------------------------------------------------------