def intYY_ey(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *
A = | | Y (u,v) sin(u)*sin(v) Y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` = 1 and `|m1-m2|` = 1.
"""
if abs(l1-l2) != 1 or abs(m1-m2) != 1:
return 0.0
scale = C(l1,m1)*C(l2,m2)*np.sign(m1-m2)*np.pi/1j
if abs(m1) == abs(m2)+1:
return scale*np.sign(l1-l2)/(2.*l2+1.)*ilegendre(l1,m1)
else:
assert abs(m1) == abs(m2)-1
return scale*np.sign(l2-l1)/(2.*l1+1.)*ilegendre(l2,m2)
def intYY_ey(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *
A = | | Y (u,v) sin(u)*sin(v) Y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` = 1 and `|m1-m2|` = 1.
"""
if abs(l1-l2) != 1 or abs(m1-m2) != 1:
return 0.0
scale = C(l1,m1)*C(l2,m2)*np.sign(m1-m2)*np.pi/1j
if abs(m1) == abs(m2)+1:
return scale*np.sign(l1-l2)/(2.*l2+1.)*ilegendre(l1,m1)
else:
assert abs(m1) == abs(m2)-1
return scale*np.sign(l2-l1)/(2.*l1+1.)*ilegendre(l2,m2)
def intYY(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ /
A = | | y (u,v) y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if l1 = l2 and m1 = m2.
"""
if (l1,m1) == (l2,m2):
return C(l1,m1)**2*2*np.pi*ilegendre(l1,m1) # == 1 always, XXX right?
else:
return 0.0
def intYY_ez(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *
A = | | Y (u,v) cos(u) Y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` = 1 and m1 = m2.
"""
if abs(l1-l2) != 1 or m1 != m2:
return 0.0
scale = C(l1,m1)*C(l2,m2)*2*np.pi
return scale*(l2-np.sign(l1-l2)*abs(m1)+heaviside(l1-l2))/(2.*l2+1.)*ilegendre(l1,m1)
def intYY_ez(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *
A = | | Y (u,v) cos(u) Y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` = 1 and m1 = m2.
"""
if abs(l1-l2) != 1 or m1 != m2:
return 0.0
scale = C(l1,m1)*C(l2,m2)*2*np.pi
return scale*(l2-np.sign(l1-l2)*abs(m1)+heaviside(l1-l2))/(2.*l2+1.)*ilegendre(l1,m1)
def intYY(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / *
A = | | Y (u,v) Y (u,v) sin(u) dv du
LL' / / lm l'm'
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if l1 = l2 and m1 = m2.
"""
if (l1,m1) == (l2,m2):
return C(l1,m1)**2*2*np.pi*ilegendre(l1,m1) # == 1 always, XXX right?
else:
return 0.0
def intYdYdtheta_ez(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / * d Y(u,v)
A = - | | Y (u,v) sin(u) --- l'm' sin(u) dv du
LL' / / lm du
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` = 1 and m1 = m2.
"""
if abs(l1 - l2) != 1 or m1 != m2:
return 0.0
scale = -C(l1, m1) * C(l2, m2) * 2 * np.pi
return scale * np.sign(l1 - l2) * (l2 + 1 - heaviside(l1 - l2)) * (
l2 - np.sign(l1 - l2) * abs(m1) +
heaviside(l1 - l2)) / (2 * l2 + 1) * ilegendre(l1, m1)
def intYdYdtheta_ez(l1, m1, l2, m2):
"""Calculates::
pi 2pi
/ / * d Y(u,v)
A = - | | Y (u,v) sin(u) --- l'm' sin(u) dv du
LL' / / lm du
0 0
where u = theta and v = phi in the usual notation. Note that the result
is only non-zero if `|l1-l2|` = 1 and m1 = m2.
"""
if abs(l1-l2) != 1 or m1 != m2:
return 0.0
scale = -C(l1,m1)*C(l2,m2)*2*np.pi
return scale*np.sign(l1-l2)*(l2+1-heaviside(l1-l2))*(l2-np.sign(l1-l2)*abs(m1)+heaviside(l1-l2))/(2*l2+1)*ilegendre(l1,m1)
