def z_zn(u, m):
r"""
A function evaluate Jacobi Zeta (zn(u,m)) function (eq 16.26.12, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi Zeta function evaluated for the argument `u` and parameter `m`
"""
phi = z_am(u, m)
zn = ellipe(phi, m) - ellipe(m) * u / ellipk(m)
return zn
def z_zn(u, m):
r"""
A function evaluate Jacobi Zeta (zn(u,m)) function (eq 16.26.12, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi Zeta function evaluated for the argument `u` and parameter `m`
"""
phi = z_am(u, m)
zn = ellipe(phi, m) - ellipe(m) * u / ellipk(m)
return zn
def calc_t_r(qr, r1, r2, r3, r4, En, Lz, aa, M=1):
"""
delta_t_r in Drasco and Hughes (2005)
Parameters:
qr (float)
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
En (float): energy
Lz (float): angular momentum
aa (float): spin
Keyword Args:
M (float): mass
Returns:
t_r (float)
"""
pi = mp.pi
psi_r = calc_psi_r(qr, r1, r2, r3, r4)
kr = ((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4))
rp = M + sqrt(-(aa**2) + M**2)
rm = M - sqrt(-(aa**2) + M**2)
hr = (r1 - r2) / (r1 - r3)
hp = ((r1 - r2) * (r3 - rp)) / ((r1 - r3) * (r2 - rp))
hm = ((r1 - r2) * (r3 - rm)) / ((r1 - r3) * (r2 - rm))
return -((En *
((-4 * (r2 - r3) *
(-(((-2 * aa**2 + (4 - (aa * Lz) / En) * rm) *
((qr * ellippi(hm, kr)) / pi - ellippi(hm, psi_r, kr))) /
((r2 - rm) * (r3 - rm))) +
((-2 * aa**2 + (4 - (aa * Lz) / En) * rp) *
((qr * ellippi(hp, kr)) / pi - ellippi(hp, psi_r, kr))) /
((r2 - rp) * (r3 - rp)))) / (-rm + rp) + 4 * (r2 - r3) *
((qr * ellippi(hr, kr)) / pi - ellippi(hr, psi_r, kr)) +
(r2 - r3) * (r1 + r2 + r3 + r4) *
((qr * ellippi(hr, kr)) / pi - ellippi(hr, psi_r, kr)) +
(r1 - r3) * (r2 - r4) *
((qr * ellipe(kr)) / pi - ellipe(psi_r, kr) +
(hr * cos(psi_r) * sin(psi_r) * sqrt(1 - kr * sin(psi_r)**2)) /
(1 - hr * sin(psi_r)**2)))) / sqrt(
(1 - En**2) * (r1 - r3) * (r2 - r4)))
def energy_density_at_origin(k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
k1 = sqrt(1-k**2)
A = 32*(k**2 *(-K**2 * k**2 +E**2-4*E*K+3* K**2 + k**2)-2*(E-K)**2)**2/(k**8 * K**4 * k1**2)
return A.real
def energy_density_at_origin(k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
k1 = sqrt(1-k**2)
A = 32*(k**2 *(-K**2 * k**2 +E**2-4*E*K+3* K**2 + k**2)-2*(E-K)**2)**2/(k**8 * K**4 * k1**2)
return A.real
def testing_components_energy_density(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3)):
return 4
if (is_awc_branch_point(k, x1, x2, x3)):
return 4
zeta = calc_zeta(k, x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x = [x1, x2, x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm = (2 * E - K) / K
k1 = sqrt(1 - k**2)
xp = x[0] + complex(0, 1) * x[1]
xm = x[0] - complex(0, 1) * x[1]
S = sqrt(K**2 - 4 * xp * xm)
SP = sqrt(K**2 - 4 * xp**2)
SM = sqrt(K**2 - 4 * xm**2)
SPM = sqrt(-k1**2 * (K**2 * k**2 - 4 * xm * xp) + (xm - xp)**2)
R = 2 * K**2 * k1**2 - S**2 - 8 * x[2]**2
RM = complex(0, 1) * SM**2 * (xm * (2 * k1**2 - 1) +
xp) - (16 * complex(0, 1)) * xm * x[2]**2
RP = complex(0, 1) * SM**2 * (xp * (2 * k1**2 - 1) +
xm) + (16 * complex(0, 1)) * xp * x[2]**2
RMBAR = -complex(0, 1) * SP**2 * (xp *
(2 * k1**2 - 1) + xm) + 16 * complex(
0, 1) * xp * x[2]**2
RPBAR = -complex(0, 1) * SP**2 * (xm *
(2 * k1**2 - 1) + xp) - 16 * complex(
0, 1) * xm * x[2]**2
r = sqrt(x[0]**2 + x[1]**2 + x[2]**2)
DM = dmus(zeta, x, k)
DZ = dzetas(zeta, x, k)
DDM = ddmus(zeta, x, k)
DDZ = ddzetas(zeta, x, k)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
# DGS1 = dgrams1(zeta, mu, DM, DZ, x, k)
#
# DGS2 = dgrams2(zeta, mu, DM, DZ, x, k)
#
# DGS3 = dgrams3(zeta, mu, DM, DZ, x, k)
# return exp(-6 * mu[0])
print zeta, '\n', abel, '\n', mu, '\n' #mu[0], abel[0], exp(-6*mu[0])
return
def calc_t_z(qz, zp, zm, En, aa):
"""
delta_t_theta in Drasco and Hughes (2003?)
Parameters:
qz (float)
zp (float): polar root
zm (float): polar root
En (float): energy
aa (float): spin
Returns:
t_z (float)
"""
psi_z = calc_psi_z(qz, zp, zm, En, aa)
ktheta = (aa**2 * (1 - En**2) * zm**2) / zp**2
return (En * zp * ((2 * (pi / 2.0 + qz) * ellipe(ktheta)) / pi -
ellipe(psi_z, ktheta))) / (1 - En**2)
def one_side(x,y):
m = ((-1 + x)*(1 + y))/((1 + x)*(-1 + y))
k = sqrt(m)
u = asin(1/k)
EE = ellipe(m)
EF = re(ellipf(u,m))
n = (-1 + x)/(-1 + y)
EPI = ellippi(n/m,1/m)/k
#EPI = ellippi(n,u,m)
return re(-(EE*(1 + x)*(-1 + y) + (x - y)*(EF + EPI*(x-y) + EF*y))/sqrt(((1 + x)*(1 - y))))
def dmus(zeta, x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DM = [None,None,None,None]
DM[0] = [None]*3
DM[1] = [None]*3
DM[2] = [None]*3
DM[3] = [None]*3
DM[0][0] = complex(0, 1) * (E * K * zeta[0] ** 2 - K ** 2 * zeta[0] ** 2 + 4 * zeta[0] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[0] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[0] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[0] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[0] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[0] - K ** 2 * zeta[0] ** 3 + 4 * x[0] ** 2 * zeta[0] + complex(0, -12) * zeta[0] ** 2 * x[0] * x[2] - 4 * zeta[0] ** 3 * x[1] ** 2 - 12 * zeta[0] ** 2 * x[1] * x[2] - K ** 2 * zeta[0] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[0] - 8 * zeta[0] * x[2] ** 2 + 4 * x[1] * x[2])
DM[1][0] = complex(0, 1) * (E * K * zeta[1] ** 2 - K ** 2 * zeta[1] ** 2 + 4 * zeta[1] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[1] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[1] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[1] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[1] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[1] - K ** 2 * zeta[1] ** 3 + 4 * x[0] ** 2 * zeta[1] + complex(0, -12) * zeta[1] ** 2 * x[0] * x[2] - 4 * zeta[1] ** 3 * x[1] ** 2 - 12 * zeta[1] ** 2 * x[1] * x[2] - K ** 2 * zeta[1] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[1] - 8 * zeta[1] * x[2] ** 2 + 4 * x[1] * x[2])
DM[2][0] = complex(0, 1) * (E * K * zeta[2] ** 2 - K ** 2 * zeta[2] ** 2 + 4 * zeta[2] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[2] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[2] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[2] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[2] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[2] - K ** 2 * zeta[2] ** 3 + 4 * x[0] ** 2 * zeta[2] + complex(0, -12) * zeta[2] ** 2 * x[0] * x[2] - 4 * zeta[2] ** 3 * x[1] ** 2 - 12 * zeta[2] ** 2 * x[1] * x[2] - K ** 2 * zeta[2] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[2] - 8 * zeta[2] * x[2] ** 2 + 4 * x[1] * x[2])
DM[3][0] = complex(0, 1) * (E * K * zeta[3] ** 2 - K ** 2 * zeta[3] ** 2 + 4 * zeta[3] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[3] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[3] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[3] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[3] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[3] - K ** 2 * zeta[3] ** 3 + 4 * x[0] ** 2 * zeta[3] + complex(0, -12) * zeta[3] ** 2 * x[0] * x[2] - 4 * zeta[3] ** 3 * x[1] ** 2 - 12 * zeta[3] ** 2 * x[1] * x[2] - K ** 2 * zeta[3] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[3] - 8 * zeta[3] * x[2] ** 2 + 4 * x[1] * x[2])
DM[0][1] = (complex(0, 4) * zeta[0] ** 2 * x[1] * x[0] + E * K * zeta[0] ** 2 + 4 * zeta[0] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[0] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[0] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[0] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[0] - K ** 2 * zeta[0] ** 3 + 4 * x[0] ** 2 * zeta[0] + complex(0, -12) * zeta[0] ** 2 * x[0] * x[2] - 4 * zeta[0] ** 3 * x[1] ** 2 - 12 * zeta[0] ** 2 * x[1] * x[2] - K ** 2 * zeta[0] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[0] - 8 * zeta[0] * x[2] ** 2 + 4 * x[1] * x[2])
DM[1][1] = (complex(0, 4) * zeta[1] ** 2 * x[1] * x[0] + E * K * zeta[1] ** 2 + 4 * zeta[1] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[1] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[1] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[1] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[1] - K ** 2 * zeta[1] ** 3 + 4 * x[0] ** 2 * zeta[1] + complex(0, -12) * zeta[1] ** 2 * x[0] * x[2] - 4 * zeta[1] ** 3 * x[1] ** 2 - 12 * zeta[1] ** 2 * x[1] * x[2] - K ** 2 * zeta[1] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[1] - 8 * zeta[1] * x[2] ** 2 + 4 * x[1] * x[2])
DM[2][1] = (complex(0, 4) * zeta[2] ** 2 * x[1] * x[0] + E * K * zeta[2] ** 2 + 4 * zeta[2] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[2] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[2] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[2] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[2] - K ** 2 * zeta[2] ** 3 + 4 * x[0] ** 2 * zeta[2] + complex(0, -12) * zeta[2] ** 2 * x[0] * x[2] - 4 * zeta[2] ** 3 * x[1] ** 2 - 12 * zeta[2] ** 2 * x[1] * x[2] - K ** 2 * zeta[2] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[2] - 8 * zeta[2] * x[2] ** 2 + 4 * x[1] * x[2])
DM[3][1] = (complex(0, 4) * zeta[3] ** 2 * x[1] * x[0] + E * K * zeta[3] ** 2 + 4 * zeta[3] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[3] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[3] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[3] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[3] - K ** 2 * zeta[3] ** 3 + 4 * x[0] ** 2 * zeta[3] + complex(0, -12) * zeta[3] ** 2 * x[0] * x[2] - 4 * zeta[3] ** 3 * x[1] ** 2 - 12 * zeta[3] ** 2 * x[1] * x[2] - K ** 2 * zeta[3] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[3] - 8 * zeta[3] * x[2] ** 2 + 4 * x[1] * x[2])
DM[0][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[0] + complex(0, 2) * zeta[0] ** 2 * x[0] * x[2] + 2 * zeta[0] ** 2 * x[1] * x[2] + E * K * zeta[0] + complex(0, 2) * x[0] * x[2] + 4 * zeta[0] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[0] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[0] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[0] - K ** 2 * zeta[0] ** 3 + 4 * x[0] ** 2 * zeta[0] + complex(0, -12) * zeta[0] ** 2 * x[0] * x[2] - 4 * zeta[0] ** 3 * x[1] ** 2 - 12 * zeta[0] ** 2 * x[1] * x[2] - K ** 2 * zeta[0] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[0] - 8 * zeta[0] * x[2] ** 2 + 4 * x[1] * x[2])
DM[1][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[1] + complex(0, 2) * zeta[1] ** 2 * x[0] * x[2] + 2 * zeta[1] ** 2 * x[1] * x[2] + E * K * zeta[1] + complex(0, 2) * x[0] * x[2] + 4 * zeta[1] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[1] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[1] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[1] - K ** 2 * zeta[1] ** 3 + 4 * x[0] ** 2 * zeta[1] + complex(0, -12) * zeta[1] ** 2 * x[0] * x[2] - 4 * zeta[1] ** 3 * x[1] ** 2 - 12 * zeta[1] ** 2 * x[1] * x[2] - K ** 2 * zeta[1] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[1] - 8 * zeta[1] * x[2] ** 2 + 4 * x[1] * x[2])
DM[2][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[2] + complex(0, 2) * zeta[2] ** 2 * x[0] * x[2] + 2 * zeta[2] ** 2 * x[1] * x[2] + E * K * zeta[2] + complex(0, 2) * x[0] * x[2] + 4 * zeta[2] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[2] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[2] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[2] - K ** 2 * zeta[2] ** 3 + 4 * x[0] ** 2 * zeta[2] + complex(0, -12) * zeta[2] ** 2 * x[0] * x[2] - 4 * zeta[2] ** 3 * x[1] ** 2 - 12 * zeta[2] ** 2 * x[1] * x[2] - K ** 2 * zeta[2] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[2] - 8 * zeta[2] * x[2] ** 2 + 4 * x[1] * x[2])
DM[3][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[3] + complex(0, 2) * zeta[3] ** 2 * x[0] * x[2] + 2 * zeta[3] ** 2 * x[1] * x[2] + E * K * zeta[3] + complex(0, 2) * x[0] * x[2] + 4 * zeta[3] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[3] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[3] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[3] - K ** 2 * zeta[3] ** 3 + 4 * x[0] ** 2 * zeta[3] + complex(0, -12) * zeta[3] ** 2 * x[0] * x[2] - 4 * zeta[3] ** 3 * x[1] ** 2 - 12 * zeta[3] ** 2 * x[1] * x[2] - K ** 2 * zeta[3] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[3] - 8 * zeta[3] * x[2] ** 2 + 4 * x[1] * x[2])
return DM
def dzetas(zeta, x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DZ = [None]*4
DZ[0] = [None]*3
DZ[1] = [None]*3
DZ[2] = [None]*3
DZ[3] = [None]*3
DZ[0][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[0] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[0] + 2 * x[2]) + K ** 2 * (4 * zeta[0] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[0]) / 4)
DZ[1][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[1] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[1] + 2 * x[2]) + K ** 2 * (4 * zeta[1] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[1]) / 4)
DZ[2][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[2] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[2] + 2 * x[2]) + K ** 2 * (4 * zeta[2] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[2]) / 4)
DZ[3][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[3] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[3] + 2 * x[2]) + K ** 2 * (4 * zeta[3] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[3]) / 4)
DZ[0][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (zeta[0] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[0] + 2 * x[2]) + K ** 2 * (4 * zeta[0] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[0]) / 4)
DZ[1][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (zeta[1] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[1] + 2 * x[2]) + K ** 2 * (4 * zeta[1] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[1]) / 4)
DZ[2][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (zeta[2] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[2] + 2 * x[2]) + K ** 2 * (4 * zeta[2] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[2]) / 4)
DZ[3][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (zeta[3] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[3] + 2 * x[2]) + K ** 2 * (4 * zeta[3] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[3]) / 4)
DZ[0][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * zeta[0] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[0] + 2 * x[2]) + K ** 2 * (4 * zeta[0] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[0]) / 4)
DZ[1][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * zeta[1] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[1] + 2 * x[2]) + K ** 2 * (4 * zeta[1] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[1]) / 4)
DZ[2][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * zeta[2] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[2] + 2 * x[2]) + K ** 2 * (4 * zeta[2] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[2]) / 4)
DZ[3][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * zeta[3] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[3] + 2 * x[2]) + K ** 2 * (4 * zeta[3] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[3]) / 4)
return DZ
def evaluate(self, time):
"""
Calculate a light curve according to the analytical models
given by Pal 2008.
Parameters
----------
time : array
An array of time points at which the light curve
shall be calculated.
.. note:: time = 0 -> Planet is exactly in the line of sight (phase = 0).
Returns
-------
Model : array
The analytical light curve is stored in the property `lightcurve`.
"""
# Translate the given parameters into an orbit and, finally,
# into a projected, normalized distance (z-parameter)
self._calcZList(time - self["T0"])
# 'W' parameters corresponding to notation in Pal '08
w = 6. - 2. * self["linLimb"] - self["quadLimb"]
w0 = (6. - 6. * self["linLimb"] - 12. * self["quadLimb"]) / w
w1 = (6. * self["linLimb"] + 12. * self["quadLimb"]) / w
w2 = 6. * self["quadLimb"] / w
# Initialize flux decrease array
df = numpy.zeros(len(time))
# Get a list of 'cases' (according to Pal '08). Depends on radius ratio and 'z'-parameter along the orbit
ca = self._selectCases()
# Loop through z-list, and calculate the light curve at each point in z (->time)
for i in self._intrans:
# Calculate the coefficients to be substituted into the Pal '08 equation
c = self._returnCoeff(ca[i].step, self._zlist[i])
# Substitute the coefficients and get 'flux decrease'
if ca[i].step != 12:
# Calculate flux decrease only if there is an occultation
if not self.useBoost:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (
c[1] + c[2] * mpmath.ellipk(c[6]**2) +
c[3] * mpmath.ellipe(c[6]**2) +
c[4] * mpmath.ellippi(c[7], c[6]**2))
else:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (
c[1] + c[2] * self.ell.ell1(c[6]) + c[3] *
self.ell.ell2(c[6]) + c[4] * self.ell.ell3(c[7], c[6]))
self.lightcurve = (1. - df) * 1. / \
(1. + self["b"]) + self["b"] / (1.0 + self["b"])
return self.lightcurve
def testing_components_energy_density(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3) ):
return 4
if (is_awc_branch_point(k, x1, x2, x3) ):
return 4
zeta = calc_zeta(k ,x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x=[x1,x2,x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DM = dmus(zeta, x, k)
DZ = dzetas(zeta, x,k)
DDM = ddmus(zeta, x, k)
DDZ = ddzetas(zeta, x,k)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
# DGS1 = dgrams1(zeta, mu, DM, DZ, x, k)
#
# DGS2 = dgrams2(zeta, mu, DM, DZ, x, k)
#
# DGS3 = dgrams3(zeta, mu, DM, DZ, x, k)
# return exp(-6 * mu[0])
print zeta, '\n', abel, '\n', mu, '\n' #mu[0], abel[0], exp(-6*mu[0])
return
def xtau(phi):
out = np.zeros((6, ))
s, c, A, B = consts(phi)
E = ellipe(asin(w * c), 1 / w)
F = ellipf(asin(w * c), 1 / w)
out[0] = -2 * A * (s**2 * B + w * s**3) # x_D1
out[1] = A * (w * E + 3 * w * c + 2 * s * c * B - 2 * w * c**3) # y_D1
out[3] = 2 * l * F / w - 3 * w * E * A - 3 * w * c * A # y_D2
out[4] = A * (B *
(2 * w**2 * c**2 - w**2 - 1) - 2 * w**3 * s**3 + 2 * w *
(w**2 - 1) * s) # x_I
out[5] = A * (w * E + 2 * w**2 * s * c * B - 2 * w**3 * c**3 + w * c *
(2 + w**2)) # y_I
return out
def evaluate(self, time):
"""
Calculate a light curve according to the analytical models
given by Pal 2008.
Parameters
----------
time : array
An array of time points at which the light curve
shall be calculated.
.. note:: time = 0 -> Planet is exactly in the line of sight (phase = 0).
Returns
-------
Model : array
The analytical light curve is stored in the property `lightcurve`.
"""
# Translate the given parameters into an orbit and, finally,
# into a projected, normalized distance (z-parameter)
self._calcZList(time - self["T0"])
# 'W' parameters corresponding to notation in Pal '08
w = 6. - 2. * self["linLimb"] - self["quadLimb"]
w0 = (6. - 6. * self["linLimb"] - 12. * self["quadLimb"]) / w
w1 = (6. * self["linLimb"] + 12. * self["quadLimb"]) / w
w2 = 6. * self["quadLimb"] / w
# Initialize flux decrease array
df = numpy.zeros(len(time))
# Get a list of 'cases' (according to Pal '08). Depends on radius ratio and 'z'-parameter along the orbit
ca = self._selectCases()
# Loop through z-list, and calculate the light curve at each point in z (->time)
for i in self._intrans:
# Calculate the coefficients to be substituted into the Pal '08 equation
c = self._returnCoeff(ca[i].step, self._zlist[i])
# Substitute the coefficients and get 'flux decrease'
if ca[i].step != 12:
# Calculate flux decrease only if there is an occultation
if not self.useBoost:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (c[1] + c[2] * mpmath.ellipk(
c[6]**2) + c[3] * mpmath.ellipe(c[6]**2) + c[4] * mpmath.ellippi(c[7], c[6]**2))
else:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (c[1] + c[2] * self.ell.ell1(
c[6]) + c[3] * self.ell.ell2(c[6]) + c[4] * self.ell.ell3(c[7], c[6]))
self.lightcurve = (1. - df) * 1. / \
(1. + self["b"]) + self["b"] / (1.0 + self["b"])
return self.lightcurve
def mino_freqs_sc(slr, ecc, x):
"""
Mino frequencies for the SC case (aa = 0)
Parameters:
slr (float): semi-latus rectum
ecc (float): eccentricity
x (float): inclincation
Returns:
ups_r (float): radial Mino frequency
ups_theta (float): polar Mino frequency
ups_phi (float): azimuthal Mino frequency
gamma (float): time Mino frequency
"""
pi = mp.pi
ups_r = (pi * sqrt(-((slr * (-6 + 2 * ecc + slr)) /
(3 + ecc**2 - slr)))) / (2 * ellipk(
(4 * ecc) / (-6 + 2 * ecc + slr)))
ups_theta = slr / sqrt(-3 - ecc**2 + slr)
ups_phi = (slr * x) / (sqrt(-3 - ecc**2 + slr) * abs(x))
gamma = (sqrt(
(-4 * ecc**2 + (-2 + slr)**2) / (slr * (-3 - ecc**2 + slr))) *
(8 + (-(((-4 + slr) * slr**2 * (-6 + 2 * ecc + slr) * ellipe(
(4 * ecc) / (-6 + 2 * ecc + slr))) / (-1 + ecc**2)) +
(slr**2 * (28 + 4 * ecc**2 - 12 * slr + slr**2) * ellipk(
(4 * ecc) / (-6 + 2 * ecc + slr))) / (-1 + ecc**2) -
(2 * (6 + 2 * ecc - slr) *
(3 + ecc**2 - slr) * slr**2 * ellippi(
(2 * ecc * (-4 + slr)) / ((1 + ecc) *
(-6 + 2 * ecc + slr)),
(4 * ecc) / (-6 + 2 * ecc + slr),
)) / ((-1 + ecc) * (1 + ecc)**2) +
(4 * (-4 + slr) * slr * (2 * (1 + ecc) * ellipk(
(4 * ecc) /
(-6 + 2 * ecc + slr)) + (-6 - 2 * ecc + slr) * ellippi(
(2 * ecc * (-4 + slr)) / ((1 + ecc) *
(-6 + 2 * ecc + slr)),
(4 * ecc) / (-6 + 2 * ecc + slr),
))) / (1 + ecc) + 2 * (-4 + slr)**2 *
((-4 + slr) * ellipk((4 * ecc) / (-6 + 2 * ecc + slr)) -
((6 + 2 * ecc - slr) * slr * ellippi(
(16 * ecc) /
(12 + 8 * ecc - 4 * ecc**2 - 8 * slr + slr**2),
(4 * ecc) / (-6 + 2 * ecc + slr),
)) / (2 + 2 * ecc - slr))) / ((-4 + slr)**2 * ellipk(
(4 * ecc) / (-6 + 2 * ecc + slr))))) / 2.0
return ups_r, ups_theta, ups_phi, gamma
def higgs_squared(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3)):
return float(255) / float(256)
if (is_awc_branch_point(k, x1, x2, x3)):
return float(255) / float(256)
zeta = calc_zeta(k, x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x = [x1, x2, x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm = (2 * E - K) / K
k1 = sqrt(1 - k**2)
xp = x[0] + complex(0, 1) * x[1]
xm = x[0] - complex(0, 1) * x[1]
S = sqrt(K**2 - 4 * xp * xm)
SP = sqrt(K**2 - 4 * xp**2)
SM = sqrt(K**2 - 4 * xm**2)
SPM = sqrt(-k1**2 * (K**2 * k**2 - 4 * xm * xp) + (xm - xp)**2)
R = 2 * K**2 * k1**2 - S**2 - 8 * x[2]**2
RM = complex(0, 1) * SM**2 * (xm * (2 * k1**2 - 1) +
xp) - (16 * complex(0, 1)) * xm * x[2]**2
RP = complex(0, 1) * SM**2 * (xp * (2 * k1**2 - 1) +
xm) + (16 * complex(0, 1)) * xp * x[2]**2
RMBAR = -complex(0, 1) * SP**2 * (xp *
(2 * k1**2 - 1) + xm) + 16 * complex(
0, 1) * xp * x[2]**2
RPBAR = -complex(0, 1) * SP**2 * (xm *
(2 * k1**2 - 1) + xp) - 16 * complex(
0, 1) * xm * x[2]**2
r = sqrt(x[0]**2 + x[1]**2 + x[2]**2)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
inv_gram = matrix(GNUM).I
higgs = phis(zeta, mu, [x1, x2, x3], k)
return -(trace(matmul(matmul(higgs, inv_gram), matmul(higgs,
inv_gram))).real) / 2
def calc_arclength_coeff(amp): # Arc length of sinusoidal surface from 0 to pi/2 (one quarter wave) # is Complete Elliptic Integral of Second Kind, E(m) # The argument m is m = -A^2, where A is amplitude of sinusoid. amplitude = float(amp) m = - amplitude*amplitude arclength = mpmath.ellipe(m) domainlength = mpmath.pi()/2 arc_coeff = float(arclength/domainlength) return arc_coeff
def higgs_squared(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3) ):
return float(255)/float(256)
if (is_awc_branch_point(k, x1, x2, x3) ):
return float(255)/float(256)
zeta = calc_zeta(k ,x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x=[x1,x2,x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
inv_gram = matrix(GNUM).I
higgs = phis(zeta, mu, [x1, x2, x3], k)
# print GNUM, inv_gram, higgs # for testing
return -(trace(matmul( matmul(higgs, inv_gram), matmul(higgs, inv_gram) )).real)/2
def loop(R, Z, I0, Ra, N_windings):
mu0 = 4.0e-7 * np.pi
alfa = R / Ra
beta = Z / Ra
gamma = Z / R
Q = (1 + alfa)**2 + beta**2
ksq = 4.0 * alfa / Q
K = mp.ellipk(ksq)
E = mp.ellipe(ksq)
B0 = mu0 / 2.0 / Ra * I0 * N_windings
Br = gamma * B0 / np.pi / np.sqrt(Q) * (E * (1 + alfa**2 + beta**2) /
(Q - 4 * alfa) - K)
Bz = B0 / np.pi / np.sqrt(Q) * (E * (1 - alfa**2 - beta**2) /
(Q - 4 * alfa) + K)
return Br, Bz
def psi_x(z, x, beta):
"""
Eq.(24) from Ref[1] with argument zeta=0 and no constant factor e*beta**2/2/rho**2.
Note that 'x' here corresponds to 'chi = x/rho',
and 'z' here corresponds to 'xi = z/2/rho' in the paper.
"""
kap = kappa(z, x, beta)
alp = alpha(z, x, beta)
arg2 = -4 * (1 + x) / x**2
T1 = (1 / fabs(x) / (1 + x) *
((2 + 2 * x + x**2) * ellipf(alp, arg2) - x**2 * ellipe(alp, arg2)))
D = kap**2 - beta**2 * (1 + x)**2 * sin(2 * alp)**2
T2 = ((kap**2 - 2 * beta**2 * (1 + x)**2 + beta**2 * (1 + x) *
(2 + 2 * x + x**2) * cos(2 * alp)) / beta / (1 + x) / D)
T3 = -kap * sin(2 * alp) / D
T4 = kap * beta**2 * (1 + x) * sin(2 * alp) * cos(2 * alp) / D
T5 = 1 / fabs(x) * ellipf(alp, arg2) # psi_phi without e/rho**2 factor
out = re((T1 + T2 + T3 + T4) - 2 / beta**2 * T5)
return out
def grams(zeta, mu, x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
return mat([[(complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[0]) - exp(-2 * mu[3])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] + complex(0, 2) * pi - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[0]) + exp(-2 * mu[2])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[3]) - exp(-2 * mu[2])),(complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[1]) - exp(-2 * mu[0])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[2]) - exp(-2 * mu[1])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[2]) + exp(-2 * mu[0]))],[(complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[0]) - exp(-2 * mu[3])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[0]) + exp(-2 * mu[2])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[3]) - exp(-2 * mu[2])),(complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[1]) - exp(-2 * mu[0])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[2]) - exp(-2 * mu[1])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] + complex(0, -2) * pi - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[2]) + exp(-2 * mu[0]))]])
def dmus(zeta, x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm = (2 * E - K) / K
k1 = sqrt(1 - k**2)
xp = x[0] + complex(0, 1) * x[1]
xm = x[0] - complex(0, 1) * x[1]
S = sqrt(K**2 - 4 * xp * xm)
SP = sqrt(K**2 - 4 * xp**2)
SM = sqrt(K**2 - 4 * xm**2)
SPM = sqrt(-k1**2 * (K**2 * k**2 - 4 * xm * xp) + (xm - xp)**2)
R = 2 * K**2 * k1**2 - S**2 - 8 * x[2]**2
RM = complex(0, 1) * SM**2 * (xm * (2 * k1**2 - 1) +
xp) - (16 * complex(0, 1)) * xm * x[2]**2
RP = complex(0, 1) * SM**2 * (xp * (2 * k1**2 - 1) +
xm) + (16 * complex(0, 1)) * xp * x[2]**2
RMBAR = -complex(0, 1) * SP**2 * (xp *
(2 * k1**2 - 1) + xm) + 16 * complex(
0, 1) * xp * x[2]**2
RPBAR = -complex(0, 1) * SP**2 * (xm *
(2 * k1**2 - 1) + xp) - 16 * complex(
0, 1) * xm * x[2]**2
r = sqrt(x[0]**2 + x[1]**2 + x[2]**2)
DM = [None, None, None, None]
DM[0] = [None] * 3
DM[1] = [None] * 3
DM[2] = [None] * 3
DM[3] = [None] * 3
DM[0][0] = complex(0, 1) * (
E * K * zeta[0]**2 - K**2 * zeta[0]**2 + 4 * zeta[0]**2 * x[0]**2 +
complex(0, -4) * zeta[0]**2 * x[1] * x[0] + E * K - K**2 + 4 * x[0]**2
+ complex(0, -8) * zeta[0] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]
) / (4 * zeta[0]**3 * x[0]**2 + complex(0, -8) * zeta[0]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[0] - K**2 * zeta[0]**3 +
4 * x[0]**2 * zeta[0] + complex(0, -12) * zeta[0]**2 * x[0] * x[2] -
4 * zeta[0]**3 * x[1]**2 - 12 * zeta[0]**2 * x[1] * x[2] -
K**2 * zeta[0] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[0] - 8 * zeta[0] * x[2]**2 + 4 * x[1] * x[2])
DM[1][0] = complex(0, 1) * (
E * K * zeta[1]**2 - K**2 * zeta[1]**2 + 4 * zeta[1]**2 * x[0]**2 +
complex(0, -4) * zeta[1]**2 * x[1] * x[0] + E * K - K**2 + 4 * x[0]**2
+ complex(0, -8) * zeta[1] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]
) / (4 * zeta[1]**3 * x[0]**2 + complex(0, -8) * zeta[1]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[1] - K**2 * zeta[1]**3 +
4 * x[0]**2 * zeta[1] + complex(0, -12) * zeta[1]**2 * x[0] * x[2] -
4 * zeta[1]**3 * x[1]**2 - 12 * zeta[1]**2 * x[1] * x[2] -
K**2 * zeta[1] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[1] - 8 * zeta[1] * x[2]**2 + 4 * x[1] * x[2])
DM[2][0] = complex(0, 1) * (
E * K * zeta[2]**2 - K**2 * zeta[2]**2 + 4 * zeta[2]**2 * x[0]**2 +
complex(0, -4) * zeta[2]**2 * x[1] * x[0] + E * K - K**2 + 4 * x[0]**2
+ complex(0, -8) * zeta[2] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]
) / (4 * zeta[2]**3 * x[0]**2 + complex(0, -8) * zeta[2]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[2] - K**2 * zeta[2]**3 +
4 * x[0]**2 * zeta[2] + complex(0, -12) * zeta[2]**2 * x[0] * x[2] -
4 * zeta[2]**3 * x[1]**2 - 12 * zeta[2]**2 * x[1] * x[2] -
K**2 * zeta[2] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[2] - 8 * zeta[2] * x[2]**2 + 4 * x[1] * x[2])
DM[3][0] = complex(0, 1) * (
E * K * zeta[3]**2 - K**2 * zeta[3]**2 + 4 * zeta[3]**2 * x[0]**2 +
complex(0, -4) * zeta[3]**2 * x[1] * x[0] + E * K - K**2 + 4 * x[0]**2
+ complex(0, -8) * zeta[3] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]
) / (4 * zeta[3]**3 * x[0]**2 + complex(0, -8) * zeta[3]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[3] - K**2 * zeta[3]**3 +
4 * x[0]**2 * zeta[3] + complex(0, -12) * zeta[3]**2 * x[0] * x[2] -
4 * zeta[3]**3 * x[1]**2 - 12 * zeta[3]**2 * x[1] * x[2] -
K**2 * zeta[3] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[3] - 8 * zeta[3] * x[2]**2 + 4 * x[1] * x[2])
DM[0][1] = (complex(0, 4) * zeta[0]**2 * x[1] * x[0] + E * K * zeta[0]**2 +
4 * zeta[0]**2 * x[1]**2 + complex(0, 4) * x[0] * x[1] +
8 * zeta[0] * x[1] * x[2] - E * K - 4 * x[1]**2) / (
4 * zeta[0]**3 * x[0]**2 + complex(0, -8) * zeta[0]**3 *
x[0] * x[1] + 2 * K**2 * k1**2 * zeta[0] -
K**2 * zeta[0]**3 + 4 * x[0]**2 * zeta[0] +
complex(0, -12) * zeta[0]**2 * x[0] * x[2] -
4 * zeta[0]**3 * x[1]**2 - 12 * zeta[0]**2 * x[1] * x[2] -
K**2 * zeta[0] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[0] - 8 * zeta[0] * x[2]**2 +
4 * x[1] * x[2])
DM[1][1] = (complex(0, 4) * zeta[1]**2 * x[1] * x[0] + E * K * zeta[1]**2 +
4 * zeta[1]**2 * x[1]**2 + complex(0, 4) * x[0] * x[1] +
8 * zeta[1] * x[1] * x[2] - E * K - 4 * x[1]**2) / (
4 * zeta[1]**3 * x[0]**2 + complex(0, -8) * zeta[1]**3 *
x[0] * x[1] + 2 * K**2 * k1**2 * zeta[1] -
K**2 * zeta[1]**3 + 4 * x[0]**2 * zeta[1] +
complex(0, -12) * zeta[1]**2 * x[0] * x[2] -
4 * zeta[1]**3 * x[1]**2 - 12 * zeta[1]**2 * x[1] * x[2] -
K**2 * zeta[1] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[1] - 8 * zeta[1] * x[2]**2 +
4 * x[1] * x[2])
DM[2][1] = (complex(0, 4) * zeta[2]**2 * x[1] * x[0] + E * K * zeta[2]**2 +
4 * zeta[2]**2 * x[1]**2 + complex(0, 4) * x[0] * x[1] +
8 * zeta[2] * x[1] * x[2] - E * K - 4 * x[1]**2) / (
4 * zeta[2]**3 * x[0]**2 + complex(0, -8) * zeta[2]**3 *
x[0] * x[1] + 2 * K**2 * k1**2 * zeta[2] -
K**2 * zeta[2]**3 + 4 * x[0]**2 * zeta[2] +
complex(0, -12) * zeta[2]**2 * x[0] * x[2] -
4 * zeta[2]**3 * x[1]**2 - 12 * zeta[2]**2 * x[1] * x[2] -
K**2 * zeta[2] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[2] - 8 * zeta[2] * x[2]**2 +
4 * x[1] * x[2])
DM[3][1] = (complex(0, 4) * zeta[3]**2 * x[1] * x[0] + E * K * zeta[3]**2 +
4 * zeta[3]**2 * x[1]**2 + complex(0, 4) * x[0] * x[1] +
8 * zeta[3] * x[1] * x[2] - E * K - 4 * x[1]**2) / (
4 * zeta[3]**3 * x[0]**2 + complex(0, -8) * zeta[3]**3 *
x[0] * x[1] + 2 * K**2 * k1**2 * zeta[3] -
K**2 * zeta[3]**3 + 4 * x[0]**2 * zeta[3] +
complex(0, -12) * zeta[3]**2 * x[0] * x[2] -
4 * zeta[3]**3 * x[1]**2 - 12 * zeta[3]**2 * x[1] * x[2] -
K**2 * zeta[3] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[3] - 8 * zeta[3] * x[2]**2 +
4 * x[1] * x[2])
DM[0][2] = 2 * (
-K**2 * k1**2 * zeta[0] + complex(0, 2) * zeta[0]**2 * x[0] * x[2] +
2 * zeta[0]**2 * x[1] * x[2] + E * K * zeta[0] +
complex(0, 2) * x[0] * x[2] + 4 * zeta[0] * x[2]**2 - 2 * x[1] * x[2]
) / (4 * zeta[0]**3 * x[0]**2 + complex(0, -8) * zeta[0]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[0] - K**2 * zeta[0]**3 +
4 * x[0]**2 * zeta[0] + complex(0, -12) * zeta[0]**2 * x[0] * x[2] -
4 * zeta[0]**3 * x[1]**2 - 12 * zeta[0]**2 * x[1] * x[2] -
K**2 * zeta[0] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[0] - 8 * zeta[0] * x[2]**2 + 4 * x[1] * x[2])
DM[1][2] = 2 * (
-K**2 * k1**2 * zeta[1] + complex(0, 2) * zeta[1]**2 * x[0] * x[2] +
2 * zeta[1]**2 * x[1] * x[2] + E * K * zeta[1] +
complex(0, 2) * x[0] * x[2] + 4 * zeta[1] * x[2]**2 - 2 * x[1] * x[2]
) / (4 * zeta[1]**3 * x[0]**2 + complex(0, -8) * zeta[1]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[1] - K**2 * zeta[1]**3 +
4 * x[0]**2 * zeta[1] + complex(0, -12) * zeta[1]**2 * x[0] * x[2] -
4 * zeta[1]**3 * x[1]**2 - 12 * zeta[1]**2 * x[1] * x[2] -
K**2 * zeta[1] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[1] - 8 * zeta[1] * x[2]**2 + 4 * x[1] * x[2])
DM[2][2] = 2 * (
-K**2 * k1**2 * zeta[2] + complex(0, 2) * zeta[2]**2 * x[0] * x[2] +
2 * zeta[2]**2 * x[1] * x[2] + E * K * zeta[2] +
complex(0, 2) * x[0] * x[2] + 4 * zeta[2] * x[2]**2 - 2 * x[1] * x[2]
) / (4 * zeta[2]**3 * x[0]**2 + complex(0, -8) * zeta[2]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[2] - K**2 * zeta[2]**3 +
4 * x[0]**2 * zeta[2] + complex(0, -12) * zeta[2]**2 * x[0] * x[2] -
4 * zeta[2]**3 * x[1]**2 - 12 * zeta[2]**2 * x[1] * x[2] -
K**2 * zeta[2] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[2] - 8 * zeta[2] * x[2]**2 + 4 * x[1] * x[2])
DM[3][2] = 2 * (
-K**2 * k1**2 * zeta[3] + complex(0, 2) * zeta[3]**2 * x[0] * x[2] +
2 * zeta[3]**2 * x[1] * x[2] + E * K * zeta[3] +
complex(0, 2) * x[0] * x[2] + 4 * zeta[3] * x[2]**2 - 2 * x[1] * x[2]
) / (4 * zeta[3]**3 * x[0]**2 + complex(0, -8) * zeta[3]**3 * x[0] * x[1] +
2 * K**2 * k1**2 * zeta[3] - K**2 * zeta[3]**3 +
4 * x[0]**2 * zeta[3] + complex(0, -12) * zeta[3]**2 * x[0] * x[2] -
4 * zeta[3]**3 * x[1]**2 - 12 * zeta[3]**2 * x[1] * x[2] -
K**2 * zeta[3] + complex(0, -4) * x[0] * x[2] +
4 * x[1]**2 * zeta[3] - 8 * zeta[3] * x[2]**2 + 4 * x[1] * x[2])
return DM
def mino_freqs_kerr(r1, r2, r3, r4, En, Lz, Q, aa, slr, ecc, x, M=1):
"""
Mino frequencies for the Kerr case (aa != 0)
Parameters:
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
En (float): energy
Lz (float): angular momentum
Q (float): Carter constant
aa (float): spin
slr (float): semi-latus rectum
ecc (float): eccentricity
x (float): inclincation
Keyword Args:
M (float): mass
Returns:
ups_r (float): radial Mino frequency
ups_theta (float): polar Mino frequency
ups_phi (float): azimuthal Mino frequency
gamma (float): time Mino frequency
"""
# En, Lz, Q = calc_eq_constants(aa, slr, ecc)
# r1, r2, r3, r4 = radial_roots(En, Q, aa, slr, ecc, M)
pi = mp.pi
L2 = Lz * Lz
aa2 = aa * aa
En2 = En * En
M2 = M * M
# polar pieces
zm = 1 - x * x
# a2zp = (L2 + aa2*(-1 + En2)*(-1 + zm))/((-1 + En2)*(-1 + zm))
eps0zp = -((L2 + aa2 * (-1 + En2) * (-1 + zm)) / (L2 * (-1 + zm)))
zmOverzp = (aa2 * (-1 + En2) * (-1 + zm) * zm) / (L2 + aa2 * (-1 + En2) *
(-1 + zm))
kr = sqrt(((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4)))
ktheta = sqrt(zmOverzp)
kr2 = kr * kr
ktheta2 = ktheta * ktheta
ellipticK_r = ellipk(kr2)
ellipticK_theta = ellipk(ktheta2)
rp = M + sqrt(M2 - aa2)
rm = M - sqrt(M2 - aa2)
hr = (r1 - r2) / (r1 - r3)
hp = ((r1 - r2) * (r3 - rp)) / ((r1 - r3) * (r2 - rp))
hm = ((r1 - r2) * (r3 - rm)) / ((r1 - r3) * (r2 - rm))
ellipticPi_hmkr = ellippi(hm, kr2)
ellipticPi_hpkr = ellippi(hp, kr2)
ellipticPi_hrkr = ellippi(hr, kr2)
ellipticPi_zmktheta = ellippi(zm, ktheta2)
ellipticE_kr = ellipe(kr2)
ellipticE_ktheta = ellipe(ktheta2)
ups_r = (pi * sqrt((1 - En2) * (r1 - r3) * (r2 - r4))) / (2 * ellipticK_r)
ups_theta = (sqrt(eps0zp) * Lz * pi) / (2.0 * ellipticK_theta)
ups_phi = (2 * aa * ups_r * (-(
((-(aa * Lz) + 2 * En * M * rm) *
(ellipticK_r - ((r2 - r3) * ellipticPi_hmkr) / (r2 - rm))) /
(r3 - rm)) + (
(-(aa * Lz) + 2 * En * M * rp) *
(ellipticK_r - ((r2 - r3) * ellipticPi_hpkr) /
(r2 - rp))) / (r3 - rp))) / (pi * sqrt(
(1 - En2) * (r1 - r3) * (r2 - r4)) * (-rm + rp)) + (
2 * ups_theta * ellipticPi_zmktheta) / (sqrt(eps0zp) * pi)
gamma = (4 * En * M2 + (2 * En * ups_theta * (L2 + aa2 * (-1 + En2) *
(-1 + zm)) *
(-ellipticE_ktheta + ellipticK_theta)) /
((-1 + En2) * sqrt(eps0zp) * Lz * pi * (-1 + zm)) +
(2 * ups_r *
((2 * M *
(-(((-2 * aa2 * En * M + (-(aa * Lz) + 4 * En * M2) * rm) *
(ellipticK_r - ((r2 - r3) * ellipticPi_hmkr) /
(r2 - rm))) / (r3 - rm)) +
((-2 * aa2 * En * M + (-(aa * Lz) + 4 * En * M2) * rp) *
(ellipticK_r - ((r2 - r3) * ellipticPi_hpkr) /
(r2 - rp))) / (r3 - rp))) / (-rm + rp) + 2 * En * M *
(r3 * ellipticK_r + (r2 - r3) * ellipticPi_hrkr) +
(En *
((r1 - r3) * (r2 - r4) * ellipticE_kr +
(-(r1 * r2) + r3 * (r1 + r2 + r3)) * ellipticK_r + (r2 - r3) *
(r1 + r2 + r3 + r4) * ellipticPi_hrkr)) / 2.0)) / (pi * sqrt(
(1 - En2) * (r1 - r3) * (r2 - r4))))
return ups_r, ups_theta, ups_phi, gamma
def f42(x):
# ellint_Ecomp
x = mpmath.mpf(x)
return mpmath.ellipe(x * x)
def f_func(r,s): alpha=0.5*(r/s+s/r) t1 = 4*np.sqrt(alpha-1)/(alpha*alpha-1)*mp.ellipe(-2/(alpha-1)) t2 = 2.0*(mp.ellipk(2./(1-alpha))/np.sqrt(alpha-1)+mp.ellipk(2/(1+alpha))/np.sqrt(alpha+1)) return -float(s*s*G/np.power(2.0*r*s,1.5)*((r/s-alpha)*t1+t2))
def get_W(self, k2, phi1, phi2):
"""
Return the vector of `W` integrals, computed
recursively given a lower boundary condition
(analytic in terms of elliptic integrals) and an upper
boundary condition (computed numerically).
The term `W_j` is the solution to the integral of
sin(u)^(2 * j) * sqrt(1 - sin(u)^2 / (1 - k^2))
from u = u1 to u = u2, where
u = (pi - 2 * phi) / 4
"""
# Useful quantities
N = 2 * self.lmax + 4
kc2 = 1 - k2
u1 = 0.25 * (np.pi - 2 * phi1)
u2 = 0.25 * (np.pi - 2 * phi2)
u = np.array([u1, u2])
sinu = np.sin(u)
cosu = np.cos(u)
diff = lambda x: np.diff(x)[0]
D2 = 1 - sinu**2 / kc2
if k2 >= 1:
sgn = 1
else:
sgn = -1
D3 = sgn * abs(D2)**1.5
# The two boundary conditions
if sgn == 1:
f0 = float((ellipe(u[1], 1 / kc2) - ellipe(u[0], 1 / kc2)).real)
else:
f0 = float((ellipe(u[1], 1 / kc2) - ellipe(u[0], 1 / kc2)).imag)
"""
For reference, this expression can be written in terms of the
real parts of elliptic integrals as follows:
n = np.floor((u + 0.5 * np.pi) / np.pi)
term = np.sqrt((sinu ** 2 / kc2 - 1) * (1 / k2 - 1))
A = n * np.pi + (-1) ** n * np.arcsin(term / sinu)
F = float((ellipf(A[1], k2) - ellipf(A[0], k2)).real)
E = float((ellipe(A[1], k2) - ellipe(A[0], k2)).real)
C = diff(
(np.sin(A) * np.cos(A)) / np.sqrt(1 - k2 * np.sin(A) ** 2)
)
f0 = -np.sqrt(kc2) * (F - E / kc2 + k2 / kc2 * C)
"""
fN = quad(
lambda u: np.sin(u)**(2 * N) * abs(1 - np.sin(u)**2 / kc2)**0.5,
u1,
u2,
epsabs=1e-12,
epsrel=1e-12,
)[0]
# The recursion coefficients
A = lambda j: 2 * (j + (j - 1) * kc2) / (2 * j + 1)
B = lambda j: -(2 * j - 3) / (2 * j + 1) * kc2
C = lambda j: diff(kc2 * sinu**(2 * j - 3) * cosu * D3) / (2 * j + 1)
# Solve the linear system
return self.solve(f0, fN, A, B, C, N)
def ellip_sec(self,k):
return mp.ellipe(k**2)
def z_zn(u, m):
"""Jacobi Zeta (zn(u,m)) function (eq 16.26.12, [Abramowitz]_)."""
phi = z_am(u, m)
zn = ellipe(phi, m) - ellipe(m) * u / ellipk(m)
return zn
def energy_density(k, x1, x2, x3): # If there is a multiple root or branch point a value of maxint-1 will be returned; maxint is globally set.
try:
if (is_awc_multiple_root(k, x1, x2, x3) ):
return -1
if (is_awc_branch_point(k, x1, x2, x3) ):
return -2
zeta = calc_zeta(k ,x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x=[x1,x2,x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DM = dmus(zeta, x, k)
DZ = dzetas(zeta, x,k)
DDM = ddmus(zeta, x, k)
DDZ = ddzetas(zeta, x,k)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
inv_gram = matrix(GNUM).I
higgs = phis(zeta, mu, [x1, x2, x3], k)
DGS1 = dgrams1(zeta, mu, DM, DZ, x, k)
DGS2 = dgrams2(zeta, mu, DM, DZ, x, k)
DGS3 = dgrams3(zeta, mu, DM, DZ, x, k)
# Using the hermiticity properties its faster to evaluate the matrix entries just once and so if we evaluate
# the *12 element the *21 is minus the conjugate of this
# DHS1 = mat([[ dphis111(zeta, mu, DM, DZ, [x1, x2, x3], k), dphis112(zeta, mu, DM, DZ, [x1, x2, x3], k)],
# [ dphis121(zeta, mu, DM, DZ, [x1, x2, x3], k), dphis122(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DH112 = dphis112(zeta, mu, DM, DZ, [x1, x2, x3], k)
DHS1 = mat([[ dphis111(zeta, mu, DM, DZ, [x1, x2, x3], k), DH112 ],
[ -conj(DH112), dphis122(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DH212 = dphis212(zeta, mu, DM, DZ, [x1, x2, x3], k)
DHS2 = mat([[ dphis211(zeta, mu, DM, DZ, [x1, x2, x3], k), DH212 ],
[ -conj(DH212), dphis222(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DH312 = dphis312(zeta, mu, DM, DZ, [x1, x2, x3], k)
DHS3 = mat([[ dphis311(zeta, mu, DM, DZ, [x1, x2, x3], k), DH312 ],
[ -conj(DH312), dphis322(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DDGS112 = ddgrams112(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDGS1 = mat([[ ddgrams111(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k), DDGS112 ],
[ -conj(DDGS112), ddgrams122(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)]])
DDGS212 = ddgrams212(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDGS2 = mat([[ ddgrams211(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k), DDGS212 ],
[ -conj(DDGS212) , ddgrams222(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)]])
DDGS312 = ddgrams312(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDGS3 = mat([[ ddgrams311(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k), DDGS312 ],
[ -conj(DDGS312), ddgrams322(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)]])
DDHS111 = ddphis111(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS112 = ddphis112(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
# DDHS121 = ddphis121(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS122 = ddphis122(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS1 = mat( [[DDHS111, DDHS112], [ -conj(DDHS112),DDHS122]])
DDHS211 = ddphis211(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS212 = ddphis212(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
# DDHS221 = ddphis221(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS222 = ddphis222(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS2 = mat( [[DDHS211, DDHS212], [-conj(DDHS212),DDHS222]])
DDHS311 = ddphis311(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS312 = ddphis312(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
# DDHS321 = ddphis321(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS322 = ddphis322(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS3 = mat( [[DDHS311, DDHS312], [-conj(DDHS312),DDHS322]])
ed1 = trace(matmul( matmul(DDHS1, inv_gram) -2* matmul( matmul(DHS1 , inv_gram), matmul(DGS1, inv_gram)) \
+ matmul(higgs, matmul( 2* matmul(matmul(inv_gram,DGS1), matmul(inv_gram, DGS1)), inv_gram) - matmul(matmul(inv_gram, DDGS1), inv_gram)),
matmul(higgs,inv_gram)) ) \
+ trace( matmul(matmul(DHS1, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS1, inv_gram)),
matmul(DHS1, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS1, inv_gram))))
ed2 = trace(matmul( matmul(DDHS2, inv_gram) -2* matmul( matmul(DHS2 , inv_gram), matmul(DGS2, inv_gram)) \
+ matmul(higgs, matmul( 2* matmul(matmul(inv_gram,DGS2), matmul(inv_gram, DGS2)), inv_gram) - matmul(matmul(inv_gram, DDGS2), inv_gram)),
matmul(higgs,inv_gram)) ) \
+ trace( matmul(matmul(DHS2, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS2, inv_gram)),
matmul(DHS2, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS2, inv_gram))))
ed3 = trace(matmul( matmul(DDHS3, inv_gram) -2* matmul( matmul(DHS3 , inv_gram), matmul(DGS3, inv_gram)) \
+ matmul(higgs, matmul( 2* matmul(matmul(inv_gram,DGS3), matmul(inv_gram, DGS3)), inv_gram) - matmul(matmul(inv_gram, DDGS3), inv_gram)),
matmul(higgs,inv_gram)) ) \
+ trace( matmul(matmul(DHS3, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS3, inv_gram)),
matmul(DHS3, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS3, inv_gram))))
# energy_density = -(ed1 + ed2 + ed3).real
return -(ed1 + ed2 + ed3).real
except:
return -3
def dexp(x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm = (2 * E - K) / K
k1 = sqrt(1 - k**2)
xp = x[0] + complex(0, 1) * x[1]
xm = x[0] - complex(0, 1) * x[1]
S = sqrt(K**2 - 4 * xp * xm)
SP = sqrt(K**2 - 4 * xp**2)
SM = sqrt(K**2 - 4 * xm**2)
SPM = sqrt(-k1**2 * (K**2 * k**2 - 4 * xm * xp) + (xm - xp)**2)
R = 2 * K**2 * k1**2 - S**2 - 8 * x[2]**2
RM = complex(0, 1) * SM**2 * (xm * (2 * k1**2 - 1) +
xp) - (16 * complex(0, 1)) * xm * x[2]**2
RP = complex(0, 1) * SM**2 * (xp * (2 * k1**2 - 1) +
xm) + (16 * complex(0, 1)) * xp * x[2]**2
RMBAR = -complex(0, 1) * SP**2 * (xp *
(2 * k1**2 - 1) + xm) + 16 * complex(
0, 1) * xp * x[2]**2
RPBAR = -complex(0, 1) * SP**2 * (xm *
(2 * k1**2 - 1) + xp) - 16 * complex(
0, 1) * xm * x[2]**2
r = sqrt(x[0]**2 + x[1]**2 + x[2]**2)
DS = [None] * 2
DSP = [None] * 2
DSM = [None] * 2
DSPM = [None] * 2
DRP = [None] * 3
DRM = [None] * 3
DRPBAR = [None] * 3
DRMBAR = [None] * 3
DS[0] = -4 * (K**2 - 4 * x[0]**2 - 4 * x[1]**2)**(-0.1e1 / 0.2e1) * x[0]
DS[1] = -4 * (K**2 - 4 * x[0]**2 - 4 * x[1]**2)**(-0.1e1 / 0.2e1) * x[1]
DSP[0] = -4 * (x[0] + complex(0, 1) * x[1]) * (
4 * x[1]**2 + complex(0, -8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.1e1 / 0.2e1)
DSP[1] = -4 * (complex(0, 1) * x[0] -
x[1]) * (4 * x[1]**2 + complex(0, -8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.1e1 / 0.2e1)
DSM[0] = 4 * (complex(0, 1) * x[1] -
x[0]) * (4 * x[1]**2 + complex(0, 8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.1e1 / 0.2e1)
DSM[1] = 4 * (x[1] + complex(0, 1) * x[0]) * (
4 * x[1]**2 + complex(0, 8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.1e1 / 0.2e1)
DSPM[0] = 4 * (-K**2 * k**2 * k1**2 + 4 * k1**2 * x[0]**2 + 4 * k1**2 *
x[1]**2 - 4 * x[1]**2)**(-0.1e1 / 0.2e1) * k1**2 * x[0]
DSPM[1] = 4 * x[1] * (
k1**2 - 1) * (-K**2 * k**2 * k1**2 + 4 * k1**2 * x[0]**2 +
4 * k1**2 * x[1]**2 - 4 * x[1]**2)**(-0.1e1 / 0.2e1)
DRP[0] = complex(
0, -8) * k1**2 * x[1]**2 - 16 * k1**2 * x[0] * x[1] + complex(
0, 2) * K**2 * k1**2 + complex(0, 16) * x[1]**2 + complex(
0, -24) * k1**2 * x[0]**2 - 16 * x[0] * x[1] + complex(
0, 16) * x[2]**2
DRP[1] = -24 * k1**2 * x[1]**2 + complex(0, -16) * k1**2 * x[0] * x[
1] - 2 * K**2 * k1**2 + 24 * x[1]**2 - 8 * k1**2 * x[0]**2 + complex(
0, 32) * x[0] * x[1] + 2 * K**2 - 8 * x[0]**2 - 16 * x[2]**2
DRP[2] = -32 * x[1] * x[2] + complex(0, 32) * x[0] * x[2]
DRM[0] = complex(
0, 24) * k1**2 * x[1]**2 - 48 * k1**2 * x[0] * x[1] + complex(
0, 2) * K**2 * k1**2 + complex(0, -16) * x[1]**2 + complex(
0, -24) * k1**2 * x[0]**2 + 16 * x[0] * x[1] + complex(
0, -16) * x[2]**2
DRM[1] = 24 * k1**2 * x[1]**2 + complex(0, 48) * k1**2 * x[0] * x[
1] + 2 * K**2 * k1**2 - 24 * x[1]**2 - 24 * k1**2 * x[0]**2 + complex(
0, -32) * x[0] * x[1] - 2 * K**2 + 8 * x[0]**2 - 16 * x[2]**2
DRM[2] = -32 * x[1] * x[2] + complex(0, -32) * x[0] * x[2]
DRPBAR[0] = complex(
0, 8) * k1**2 * x[1]**2 - 16 * k1**2 * x[0] * x[1] + complex(
0, -2) * K**2 * k1**2 + complex(0, -16) * x[1]**2 + complex(
0, 24) * k1**2 * x[0]**2 - 16 * x[0] * x[1] + complex(
0, -16) * x[2]**2
DRPBAR[1] = -24 * k1**2 * x[1]**2 + complex(0, 16) * k1**2 * x[0] * x[
1] - 2 * K**2 * k1**2 + 24 * x[1]**2 - 8 * k1**2 * x[0]**2 + complex(
0, -32) * x[0] * x[1] + 2 * K**2 - 8 * x[0]**2 - 16 * x[2]**2
DRPBAR[2] = -32 * x[1] * x[2] + complex(0, -32) * x[0] * x[2]
DRMBAR[0] = complex(
0, -24) * k1**2 * x[1]**2 - 48 * k1**2 * x[0] * x[1] + complex(
0, -2) * K**2 * k1**2 + complex(0, 16) * x[1]**2 + complex(
0, 24) * k1**2 * x[0]**2 + 16 * x[0] * x[1] + complex(
0, 16) * x[2]**2
DRMBAR[1] = 24 * k1**2 * x[1]**2 + complex(0, -48) * k1**2 * x[0] * x[
1] + 2 * K**2 * k1**2 - 24 * x[1]**2 - 24 * k1**2 * x[0]**2 + complex(
0, 32) * x[0] * x[1] - 2 * K**2 + 8 * x[0]**2 - 16 * x[2]**2
DRMBAR[2] = -32 * x[1] * x[2] + complex(0, 32) * x[0] * x[2]
return DS, DSP, DSM, DSPM, DRP, DRM, DRPBAR, DRMBAR
def energy_density(k, x1, x2, x3): # If there is a multiple root or branch point a value of maxint-1 will be returned; maxint is globally set.
try:
if (is_awc_multiple_root(k, x1, x2, x3) ):
return -1
if (is_awc_branch_point(k, x1, x2, x3) ):
return -2
zeta = calc_zeta(k ,x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x=[x1,x2,x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DM = dmus(zeta, x, k)
DZ = dzetas(zeta, x,k)
DDM = ddmus(zeta, x, k)
DDZ = ddzetas(zeta, x,k)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
inv_gram = matrix(GNUM).I
higgs = phis(zeta, mu, [x1, x2, x3], k)
DGS1 = dgrams1(zeta, mu, DM, DZ, x, k)
DGS2 = dgrams2(zeta, mu, DM, DZ, x, k)
DGS3 = dgrams3(zeta, mu, DM, DZ, x, k)
# Using the hermiticity properties its faster to evaluate the matrix entries just once and so if we evaluate
# the *12 element the *21 is minus the conjugate of this
# DHS1 = mat([[ dphis111(zeta, mu, DM, DZ, [x1, x2, x3], k), dphis112(zeta, mu, DM, DZ, [x1, x2, x3], k)],
# [ dphis121(zeta, mu, DM, DZ, [x1, x2, x3], k), dphis122(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DH112 = dphis112(zeta, mu, DM, DZ, [x1, x2, x3], k)
DHS1 = mat([[ dphis111(zeta, mu, DM, DZ, [x1, x2, x3], k), DH112 ],
[ -conj(DH112), dphis122(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DH212 = dphis212(zeta, mu, DM, DZ, [x1, x2, x3], k)
DHS2 = mat([[ dphis211(zeta, mu, DM, DZ, [x1, x2, x3], k), DH212 ],
[ -conj(DH212), dphis222(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DH312 = dphis312(zeta, mu, DM, DZ, [x1, x2, x3], k)
DHS3 = mat([[ dphis311(zeta, mu, DM, DZ, [x1, x2, x3], k), DH312 ],
[ -conj(DH312), dphis322(zeta, mu, DM, DZ, [x1, x2, x3], k)]])
DDGS112 = ddgrams112(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDGS1 = mat([[ ddgrams111(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k), DDGS112 ],
[ -conj(DDGS112), ddgrams122(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)]])
DDGS212 = ddgrams212(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDGS2 = mat([[ ddgrams211(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k), DDGS212 ],
[ -conj(DDGS212) , ddgrams222(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)]])
DDGS312 = ddgrams312(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDGS3 = mat([[ ddgrams311(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k), DDGS312 ],
[ -conj(DDGS312), ddgrams322(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)]])
DDHS111 = ddphis111(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS112 = ddphis112(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
# DDHS121 = ddphis121(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS122 = ddphis122(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS1 = mat( [[DDHS111, DDHS112], [ -conj(DDHS112),DDHS122]])
DDHS211 = ddphis211(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS212 = ddphis212(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
# DDHS221 = ddphis221(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS222 = ddphis222(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS2 = mat( [[DDHS211, DDHS212], [-conj(DDHS212),DDHS222]])
DDHS311 = ddphis311(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS312 = ddphis312(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
# DDHS321 = ddphis321(zeta, mu,DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS322 = ddphis322(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
DDHS3 = mat( [[DDHS311, DDHS312], [-conj(DDHS312),DDHS322]])
ed1 = trace(matmul( matmul(DDHS1, inv_gram) -2* matmul( matmul(DHS1 , inv_gram), matmul(DGS1, inv_gram)) \
+ matmul(higgs, matmul( 2* matmul(matmul(inv_gram,DGS1), matmul(inv_gram, DGS1)), inv_gram) - matmul(matmul(inv_gram, DDGS1), inv_gram)),
matmul(higgs,inv_gram)) ) \
+ trace( matmul(matmul(DHS1, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS1, inv_gram)),
matmul(DHS1, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS1, inv_gram))))
ed2 = trace(matmul( matmul(DDHS2, inv_gram) -2* matmul( matmul(DHS2 , inv_gram), matmul(DGS2, inv_gram)) \
+ matmul(higgs, matmul( 2* matmul(matmul(inv_gram,DGS2), matmul(inv_gram, DGS2)), inv_gram) - matmul(matmul(inv_gram, DDGS2), inv_gram)),
matmul(higgs,inv_gram)) ) \
+ trace( matmul(matmul(DHS2, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS2, inv_gram)),
matmul(DHS2, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS2, inv_gram))))
ed3 = trace(matmul( matmul(DDHS3, inv_gram) -2* matmul( matmul(DHS3 , inv_gram), matmul(DGS3, inv_gram)) \
+ matmul(higgs, matmul( 2* matmul(matmul(inv_gram,DGS3), matmul(inv_gram, DGS3)), inv_gram) - matmul(matmul(inv_gram, DDGS3), inv_gram)),
matmul(higgs,inv_gram)) ) \
+ trace( matmul(matmul(DHS3, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS3, inv_gram)),
matmul(DHS3, inv_gram) - matmul(matmul(higgs, inv_gram), matmul(DGS3, inv_gram))))
# energy_density = -(ed1 + ed2 + ed3).real
return -(ed1 + ed2 + ed3).real
except:
return -3
def ddexp(x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DDS = [None]*2
DDSP = [None]*2
DDSM = [None]*2
DDSPM = [None]*2
DDRP = [None]*3
DDRM = [None]*3
DDRPBAR = [None]*3
DDRMBAR = [None]*3
DDS[0] = -4 * (K ** 2 - 4 * x[1] ** 2) * (K ** 2 - 4 * x[0] ** 2 - 4 * x[1] ** 2) ** (-0.3e1 / 0.2e1)
DDS[1] = -4 * (K ** 2 - 4 * x[0] ** 2) * (K ** 2 - 4 * x[0] ** 2 - 4 * x[1] ** 2) ** (-0.3e1 / 0.2e1)
DDSP[0] = -4 * K ** 2 * (4 * x[1] ** 2 + complex(0, -8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.3e1 / 0.2e1)
DDSP[1] = 4 * K ** 2 * (4 * x[1] ** 2 + complex(0, -8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.3e1 / 0.2e1)
DDSM[0] = -4 * K ** 2 * (4 * x[1] ** 2 + complex(0, 8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.3e1 / 0.2e1)
DDSM[1] = 4 * K ** 2 * (4 * x[1] ** 2 + complex(0, 8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.3e1 / 0.2e1)
DDSPM[0] = -4 * k1 ** 2 * (K ** 2 * k ** 2 * k1 ** 2 - 4 * k1 ** 2 * x[1] ** 2 + 4 * x[1] ** 2) * (-K ** 2 * k ** 2 * k1 ** 2 + 4 * k1 ** 2 * x[0] ** 2 + 4 * k1 ** 2 * x[1] ** 2 - 4 * x[1] ** 2) ** (-0.3e1 / 0.2e1)
DDSPM[1] = -4 * (k1 ** 2 - 1) * k1 ** 2 * (K ** 2 * k ** 2 - 4 * x[0] ** 2) * (-K ** 2 * k ** 2 * k1 ** 2 + 4 * k1 ** 2 * x[0] ** 2 + 4 * k1 ** 2 * x[1] ** 2 - 4 * x[1] ** 2) ** (-0.3e1 / 0.2e1)
DDRP[0] = -16 * k1 ** 2 * x[1] + complex(0, -48) * k1 ** 2 * x[0] - 16 * x[1]
DDRP[1] = -48 * k1 ** 2 * x[1] + complex(0, -16) * k1 ** 2 * x[0] + 48 * x[1] + complex(0, 32) * x[0]
DDRP[2] = -32 * x[1] + complex(0, 32) * x[0]
DDRM[0] = -48 * k1 ** 2 * x[1] + complex(0, -48) * k1 ** 2 * x[0] + 16 * x[1]
DDRM[1] = 48 * k1 ** 2 * x[1] + complex(0, 48) * k1 ** 2 * x[0] - 48 * x[1] + complex(0, -32) * x[0]
DDRM[2] = -32 * x[1] + complex(0, -32) * x[0]
DDRPBAR[0] = -16 * k1 ** 2 * x[1] + complex(0, 48) * k1 ** 2 * x[0] - 16 * x[1]
DDRPBAR[1] = -48 * k1 ** 2 * x[1] + complex(0, 16) * k1 ** 2 * x[0] + 48 * x[1] + complex(0, -32) * x[0]
DDRPBAR[2] = -32 * x[1] + complex(0, -32) * x[0]
DDRMBAR[0] = -48 * k1 ** 2 * x[1] + complex(0, 48) * k1 ** 2 * x[0] + 16 * x[1]
DDRMBAR[1] = 48 * k1 ** 2 * x[1] + complex(0, -48) * k1 ** 2 * x[0] - 48 * x[1] + complex(0, 32) * x[0]
DDRMBAR[2] = -32 * x[1] + complex(0, 32) * x[0]
def ddexp(x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm = (2 * E - K) / K
k1 = sqrt(1 - k**2)
xp = x[0] + complex(0, 1) * x[1]
xm = x[0] - complex(0, 1) * x[1]
S = sqrt(K**2 - 4 * xp * xm)
SP = sqrt(K**2 - 4 * xp**2)
SM = sqrt(K**2 - 4 * xm**2)
SPM = sqrt(-k1**2 * (K**2 * k**2 - 4 * xm * xp) + (xm - xp)**2)
R = 2 * K**2 * k1**2 - S**2 - 8 * x[2]**2
RM = complex(0, 1) * SM**2 * (xm * (2 * k1**2 - 1) +
xp) - (16 * complex(0, 1)) * xm * x[2]**2
RP = complex(0, 1) * SM**2 * (xp * (2 * k1**2 - 1) +
xm) + (16 * complex(0, 1)) * xp * x[2]**2
RMBAR = -complex(0, 1) * SP**2 * (xp *
(2 * k1**2 - 1) + xm) + 16 * complex(
0, 1) * xp * x[2]**2
RPBAR = -complex(0, 1) * SP**2 * (xm *
(2 * k1**2 - 1) + xp) - 16 * complex(
0, 1) * xm * x[2]**2
r = sqrt(x[0]**2 + x[1]**2 + x[2]**2)
DDS = [None] * 2
DDSP = [None] * 2
DDSM = [None] * 2
DDSPM = [None] * 2
DDRP = [None] * 3
DDRM = [None] * 3
DDRPBAR = [None] * 3
DDRMBAR = [None] * 3
DDS[0] = -4 * (K**2 - 4 * x[1]**2) * (K**2 - 4 * x[0]**2 -
4 * x[1]**2)**(-0.3e1 / 0.2e1)
DDS[1] = -4 * (K**2 - 4 * x[0]**2) * (K**2 - 4 * x[0]**2 -
4 * x[1]**2)**(-0.3e1 / 0.2e1)
DDSP[0] = -4 * K**2 * (4 * x[1]**2 + complex(0, -8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.3e1 / 0.2e1)
DDSP[1] = 4 * K**2 * (4 * x[1]**2 + complex(0, -8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.3e1 / 0.2e1)
DDSM[0] = -4 * K**2 * (4 * x[1]**2 + complex(0, 8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.3e1 / 0.2e1)
DDSM[1] = 4 * K**2 * (4 * x[1]**2 + complex(0, 8) * x[0] * x[1] + K**2 -
4 * x[0]**2)**(-0.3e1 / 0.2e1)
DDSPM[0] = -4 * k1**2 * (
K**2 * k**2 * k1**2 - 4 * k1**2 * x[1]**2 +
4 * x[1]**2) * (-K**2 * k**2 * k1**2 + 4 * k1**2 * x[0]**2 +
4 * k1**2 * x[1]**2 - 4 * x[1]**2)**(-0.3e1 / 0.2e1)
DDSPM[1] = -4 * (k1**2 - 1) * k1**2 * (K**2 * k**2 - 4 * x[0]**2) * (
-K**2 * k**2 * k1**2 + 4 * k1**2 * x[0]**2 + 4 * k1**2 * x[1]**2 -
4 * x[1]**2)**(-0.3e1 / 0.2e1)
DDRP[0] = -16 * k1**2 * x[1] + complex(0, -48) * k1**2 * x[0] - 16 * x[1]
DDRP[1] = -48 * k1**2 * x[1] + complex(
0, -16) * k1**2 * x[0] + 48 * x[1] + complex(0, 32) * x[0]
DDRP[2] = -32 * x[1] + complex(0, 32) * x[0]
DDRM[0] = -48 * k1**2 * x[1] + complex(0, -48) * k1**2 * x[0] + 16 * x[1]
DDRM[1] = 48 * k1**2 * x[1] + complex(
0, 48) * k1**2 * x[0] - 48 * x[1] + complex(0, -32) * x[0]
DDRM[2] = -32 * x[1] + complex(0, -32) * x[0]
DDRPBAR[0] = -16 * k1**2 * x[1] + complex(0, 48) * k1**2 * x[0] - 16 * x[1]
DDRPBAR[1] = -48 * k1**2 * x[1] + complex(
0, 16) * k1**2 * x[0] + 48 * x[1] + complex(0, -32) * x[0]
DDRPBAR[2] = -32 * x[1] + complex(0, -32) * x[0]
DDRMBAR[0] = -48 * k1**2 * x[1] + complex(0, 48) * k1**2 * x[0] + 16 * x[1]
DDRMBAR[1] = 48 * k1**2 * x[1] + complex(
0, -48) * k1**2 * x[0] - 48 * x[1] + complex(0, 32) * x[0]
DDRMBAR[2] = -32 * x[1] + complex(0, 32) * x[0]
def z_zn(u, m):
"""Jacobi Zeta (zn(u,m)) function (eq 16.26.12, [Abramowitz]_)."""
phi = z_am(u, m)
zn = ellipe(phi, m) - ellipe(m) * u / ellipk(m)
return zn
def dexp(x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DS = [None]*2
DSP = [None]*2
DSM = [None]*2
DSPM = [None]*2
DRP = [None]*3
DRM = [None]*3
DRPBAR = [None]*3
DRMBAR = [None]*3
DS[0] = -4 * (K ** 2 - 4 * x[0] ** 2 - 4 * x[1] ** 2) ** (-0.1e1 / 0.2e1) * x[0]
DS[1] = -4 * (K ** 2 - 4 * x[0] ** 2 - 4 * x[1] ** 2) ** (-0.1e1 / 0.2e1) * x[1]
DSP[0] = -4 * (x[0] + complex(0, 1) * x[1]) * (4 * x[1] ** 2 + complex(0, -8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.1e1 / 0.2e1)
DSP[1] = -4 * (complex(0, 1) * x[0] - x[1]) * (4 * x[1] ** 2 + complex(0, -8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.1e1 / 0.2e1)
DSM[0] = 4 * (complex(0, 1) * x[1] - x[0]) * (4 * x[1] ** 2 + complex(0, 8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.1e1 / 0.2e1)
DSM[1] = 4 * (x[1] + complex(0, 1) * x[0]) * (4 * x[1] ** 2 + complex(0, 8) * x[0] * x[1] + K ** 2 - 4 * x[0] ** 2) ** (-0.1e1 / 0.2e1)
DSPM[0] = 4 * (-K ** 2 * k ** 2 * k1 ** 2 + 4 * k1 ** 2 * x[0] ** 2 + 4 * k1 ** 2 * x[1] ** 2 - 4 * x[1] ** 2) ** (-0.1e1 / 0.2e1) * k1 ** 2 * x[0]
DSPM[1] = 4 * x[1] * (k1 ** 2 - 1) * (-K ** 2 * k ** 2 * k1 ** 2 + 4 * k1 ** 2 * x[0] ** 2 + 4 * k1 ** 2 * x[1] ** 2 - 4 * x[1] ** 2) ** (-0.1e1 / 0.2e1)
DRP[0] = complex(0, -8) * k1 ** 2 * x[1] ** 2 - 16 * k1 ** 2 * x[0] * x[1] + complex(0, 2) * K ** 2 * k1 ** 2 + complex(0, 16) * x[1] ** 2 + complex(0, -24) * k1 ** 2 * x[0] ** 2 - 16 * x[0] * x[1] + complex(0, 16) * x[2] ** 2
DRP[1] = -24 * k1 ** 2 * x[1] ** 2 + complex(0, -16) * k1 ** 2 * x[0] * x[1] - 2 * K ** 2 * k1 ** 2 + 24 * x[1] ** 2 - 8 * k1 ** 2 * x[0] ** 2 + complex(0, 32) * x[0] * x[1] + 2 * K ** 2 - 8 * x[0] ** 2 - 16 * x[2] ** 2
DRP[2] = -32 * x[1] * x[2] + complex(0, 32) * x[0] * x[2]
DRM[0] = complex(0, 24) * k1 ** 2 * x[1] ** 2 - 48 * k1 ** 2 * x[0] * x[1] + complex(0, 2) * K ** 2 * k1 ** 2 + complex(0, -16) * x[1] ** 2 + complex(0, -24) * k1 ** 2 * x[0] ** 2 + 16 * x[0] * x[1] + complex(0, -16) * x[2] ** 2
DRM[1] = 24 * k1 ** 2 * x[1] ** 2 + complex(0, 48) * k1 ** 2 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 - 24 * x[1] ** 2 - 24 * k1 ** 2 * x[0] ** 2 + complex(0, -32) * x[0] * x[1] - 2 * K ** 2 + 8 * x[0] ** 2 - 16 * x[2] ** 2
DRM[2] = -32 * x[1] * x[2] + complex(0, -32) * x[0] * x[2]
DRPBAR[0] = complex(0, 8) * k1 ** 2 * x[1] ** 2 - 16 * k1 ** 2 * x[0] * x[1] + complex(0, -2) * K ** 2 * k1 ** 2 + complex(0, -16) * x[1] ** 2 + complex(0, 24) * k1 ** 2 * x[0] ** 2 - 16 * x[0] * x[1] + complex(0, -16) * x[2] ** 2
DRPBAR[1] = -24 * k1 ** 2 * x[1] ** 2 + complex(0, 16) * k1 ** 2 * x[0] * x[1] - 2 * K ** 2 * k1 ** 2 + 24 * x[1] ** 2 - 8 * k1 ** 2 * x[0] ** 2 + complex(0, -32) * x[0] * x[1] + 2 * K ** 2 - 8 * x[0] ** 2 - 16 * x[2] ** 2
DRPBAR[2] = -32 * x[1] * x[2] + complex(0, -32) * x[0] * x[2]
DRMBAR[0] = complex(0, -24) * k1 ** 2 * x[1] ** 2 - 48 * k1 ** 2 * x[0] * x[1] + complex(0, -2) * K ** 2 * k1 ** 2 + complex(0, 16) * x[1] ** 2 + complex(0, 24) * k1 ** 2 * x[0] ** 2 + 16 * x[0] * x[1] + complex(0, 16) * x[2] ** 2
DRMBAR[1] = 24 * k1 ** 2 * x[1] ** 2 + complex(0, -48) * k1 ** 2 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 - 24 * x[1] ** 2 - 24 * k1 ** 2 * x[0] ** 2 + complex(0, 32) * x[0] * x[1] - 2 * K ** 2 + 8 * x[0] ** 2 - 16 * x[2] ** 2
DRMBAR[2] = -32 * x[1] * x[2] + complex(0, 32) * x[0] * x[2]
return DS, DSP, DSM, DSPM, DRP, DRM, DRPBAR, DRMBAR
def ellip(bo, ro, kappa):
# Helper variables
k2 = (1 - ro**2 - bo**2 + 2 * bo * ro) / (4 * bo * ro)
if np.abs(1 - k2) < STARRY_K2_ONE_TOL:
if k2 == 1.0:
k2 = 1 + STARRY_K2_ONE_TOL
elif k2 < 1.0:
k2 = 1.0 - STARRY_K2_ONE_TOL
else:
k2 = 1.0 + STARRY_K2_ONE_TOL
k = np.sqrt(k2)
k2inv = 1 / k2
kinv = np.sqrt(k2inv)
kc2 = 1 - k2
# Complete elliptic integrals (we'll need them to compute offsets below)
if k2 < 1:
K0 = float(ellipk(k2))
E0 = float(ellipe(k2))
E0 = np.sqrt(k2inv) * (E0 - (1 - k2) * K0)
K0 *= np.sqrt(k2)
RJ0 = 0.0
else:
K0 = float(ellipk(k2inv))
E0 = float(ellipe(k2inv))
if (bo != 0) and (bo != ro):
p0 = (ro * ro + bo * bo + 2 * ro * bo) / (ro * ro + bo * bo -
2 * ro * bo)
PI0 = float(ellippi(1 - p0, k2inv))
RJ0 = -12.0 / (1 - p0) * (PI0 - K0)
else:
RJ0 = 0.0
if k2 < 1:
# Analytic continuation from (17.4.15-16) in Abramowitz & Stegun
# A better format is here: https://dlmf.nist.gov/19.7#ii
# Helper variables
arg = kinv * np.sin(kappa / 2)
tanphi = arg / np.sqrt(1 - arg**2)
tanphi[arg >= 1] = STARRY_HUGE_TAN
tanphi[arg <= -1] = -STARRY_HUGE_TAN
# Compute the elliptic integrals
F = EllipF(tanphi, k2) * k
E = kinv * (EllipE(tanphi, k2) - kc2 * kinv * F)
# Add offsets to account for the limited domain of `el2`
for i in range(len(kappa)):
if kappa[i] > 3 * np.pi:
F[i] += 4 * K0
E[i] += 4 * E0
elif kappa[i] > np.pi:
F[i] = 2 * K0 - F[i]
E[i] = 2 * E0 - E[i]
else:
# Helper variables
tanphi = np.tan(kappa / 2)
# Compute the elliptic integrals
F = EllipF(tanphi, k2inv) # el2(tanphi, kcinv, 1, 1)
E = EllipE(tanphi, k2inv) # el2(tanphi, kcinv, 1, kc2inv)
# Add offsets to account for the limited domain of `el2`
for i in range(len(kappa)):
if kappa[i] > 3 * np.pi:
F[i] += 4 * K0
E[i] += 4 * E0
elif kappa[i] > np.pi:
F[i] += 2 * K0
E[i] += 2 * E0
# Must compute RJ separately
if np.abs(bo - ro) > STARRY_PAL_BO_EQUALS_RO_TOL:
p = (ro * ro + bo * bo -
2 * ro * bo * np.cos(kappa)) / (ro * ro + bo * bo - 2 * ro * bo)
RJ = EllipJ(kappa, k2, p)
# Add offsets to account for the limited domain of `rj`
if RJ0 != 0.0:
for i in range(len(kappa)):
if kappa[i] > 3 * np.pi:
RJ[i] += 2 * RJ0
elif kappa[i] > np.pi:
RJ[i] += RJ0
else:
RJ = np.zeros_like(kappa)
# Compute the *definite* elliptic integrals
F = pairdiff(F)
E = pairdiff(E)
PIprime = pairdiff(RJ)
return F, E, PIprime