static double jnt(double n, double x)
{
double z, zz, z3;
double cbn, n23, cbtwo;
double ai, aip, bi, bip; /* Airy functions */
double nk, fk, gk, pp, qq;
double F[5], G[4];
int k;
cbn = cbrt(n);
z = (x - n) / cbn;
cbtwo = cbrt(2.0);
/* Airy function */
zz = -cbtwo * z;
airy(zz, &ai, &aip, &bi, &bip);
/* polynomials in expansion */
zz = z * z;
z3 = zz * z;
F[0] = 1.0;
F[1] = -z / 5.0;
F[2] = polevl(z3, PF2, 1) * zz;
F[3] = polevl(z3, PF3, 2);
F[4] = polevl(z3, PF4, 3) * z;
G[0] = 0.3 * zz;
G[1] = polevl(z3, PG1, 1);
G[2] = polevl(z3, PG2, 2) * z;
G[3] = polevl(z3, PG3, 2) * zz;
#if CEPHES_DEBUG
for (k = 0; k <= 4; k++)
printf("F[%d] = %.5E\n", k, F[k]);
for (k = 0; k <= 3; k++)
printf("G[%d] = %.5E\n", k, G[k]);
#endif
pp = 0.0;
qq = 0.0;
nk = 1.0;
n23 = cbrt(n * n);
for (k = 0; k <= 4; k++) {
fk = F[k] * nk;
pp += fk;
if (k != 4) {
gk = G[k] * nk;
qq += gk;
}
#if CEPHES_DEBUG
printf("fk[%d] %.5E, gk[%d] %.5E\n", k, fk, k, gk);
#endif
nk /= n23;
}
fk = cbtwo * ai * pp / cbn + cbrt(4.0) * aip * qq / n;
return (fk);
}
void wairy( double x, dcomplex *w1, dcomplex *w2, dcomplex *w1p, dcomplex *w2p ) {
double factor = SQRTPI;
double ai, bi, aip, bip;
airy( x, &ai, &bi, &aip, &bip );
*w1 = mk_dc( ai, bi );
*w1 = rmul_dc( factor, *w1 );
*w2 = mk_dc( ai, -bi );
*w2 = rmul_dc( factor, *w2 );
*w1p = mk_dc( aip, bip );
*w1p = rmul_dc( factor, *w1p );
*w2p = mk_dc( aip, -bip );
*w2p = rmul_dc( factor, *w2p );
}
static double jnx(double n, double x)
{
double zeta, sqz, zz, zp, np;
double cbn, n23, t, z, sz;
double pp, qq, z32i, zzi;
double ak, bk, akl, bkl;
int sign, doa, dob, nflg, k, s, tk, tkp1, m;
static double u[8];
static double ai, aip, bi, bip;
/* Test for x very close to n. Use expansion for transition region if so. */
cbn = cbrt(n);
z = (x - n) / cbn;
if (fabs(z) <= 0.7)
return (jnt(n, x));
z = x / n;
zz = 1.0 - z * z;
if (zz == 0.0)
return (0.0);
if (zz > 0.0) {
sz = sqrt(zz);
t = 1.5 * (log((1.0 + sz) / z) - sz); /* zeta ** 3/2 */
zeta = cbrt(t * t);
nflg = 1;
}
else {
sz = sqrt(-zz);
t = 1.5 * (sz - acos(1.0 / z));
zeta = -cbrt(t * t);
nflg = -1;
}
z32i = fabs(1.0 / t);
sqz = cbrt(t);
/* Airy function */
n23 = cbrt(n * n);
t = n23 * zeta;
#if CEPHES_DEBUG
printf("zeta %.5E, Airy(%.5E)\n", zeta, t);
#endif
airy(t, &ai, &aip, &bi, &bip);
/* polynomials in expansion */
u[0] = 1.0;
zzi = 1.0 / zz;
u[1] = polevl(zzi, P1, 1) / sz;
u[2] = polevl(zzi, P2, 2) / zz;
u[3] = polevl(zzi, P3, 3) / (sz * zz);
pp = zz * zz;
u[4] = polevl(zzi, P4, 4) / pp;
u[5] = polevl(zzi, P5, 5) / (pp * sz);
pp *= zz;
u[6] = polevl(zzi, P6, 6) / pp;
u[7] = polevl(zzi, P7, 7) / (pp * sz);
#if CEPHES_DEBUG
for (k = 0; k <= 7; k++)
printf("u[%d] = %.5E\n", k, u[k]);
#endif
pp = 0.0;
qq = 0.0;
np = 1.0;
/* flags to stop when terms get larger */
doa = 1;
dob = 1;
akl = NPY_INFINITY;
bkl = NPY_INFINITY;
for (k = 0; k <= 3; k++) {
tk = 2 * k;
tkp1 = tk + 1;
zp = 1.0;
ak = 0.0;
bk = 0.0;
for (s = 0; s <= tk; s++) {
if (doa) {
if ((s & 3) > 1)
sign = nflg;
else
sign = 1;
ak += sign * mu[s] * zp * u[tk - s];
}
if (dob) {
m = tkp1 - s;
if (((m + 1) & 3) > 1)
sign = nflg;
else
sign = 1;
bk += sign * lambda[s] * zp * u[m];
}
zp *= z32i;
}
if (doa) {
ak *= np;
t = fabs(ak);
if (t < akl) {
akl = t;
pp += ak;
}
else
doa = 0;
}
if (dob) {
bk += lambda[tkp1] * zp * u[0];
bk *= -np / sqz;
t = fabs(bk);
if (t < bkl) {
bkl = t;
qq += bk;
}
else
dob = 0;
}
#if CEPHES_DEBUG
printf("a[%d] %.5E, b[%d] %.5E\n", k, ak, k, bk);
#endif
if (np < MACHEP)
break;
np /= n * n;
}
/* normalizing factor ( 4*zeta/(1 - z**2) )**1/4 */
t = 4.0 * zeta / zz;
t = sqrt(sqrt(t));
t *= ai * pp / cbrt(n) + aip * qq / (n23 * n);
return (t);
}
