def __init__(self, ell=2, m=0):
self.ell = ell
self.m = m
x = sym.Symbol('x')
y = sym.Symbol('y')
z = sym.Symbol('z')
r = sym.Symbol('r', real=True, positive=True)
theta = sym.Symbol('theta', real=True, positive=True)
phi = sym.Symbol('phi', real=True, positive=True)
def c(ell, m):
if (m >= 0):
return (factorial(ell - 1) *
(-2)**Abs(m)) / factorial(ell + Abs(m)) * cos(m * phi)
else:
return (factorial(ell - 1) *
(-2)**Abs(m)) / factorial(ell + Abs(m)) * sin(
Abs(m) * phi)
legendre_ell = ell + 1
Sigma = c(legendre_ell, m) * r**legendre_ell * assoc_legendre(
legendre_ell, Abs(m), cos(theta))
Sigma = sym.simplify(Sigma)
Sigma = Sigma.subs({Abs(sin(theta)): sin(theta)})
Sigma = Sigma.expand(trig=True)
self.Sigma_spherical = Sigma
Sigma = Sigma.subs({
r * cos(theta): z,
r * sin(theta) * sin(phi): y,
r * sin(theta) * cos(phi): x
})
Sigma = Sigma.subs({r**2 * sin(theta)**2: x**2 + y**2})
Sigma = Sigma.subs({r**2: x**2 + y**2 + z**2})
Sigma = sym.simplify(Sigma.expand())
self.Sigma = Sigma
Pix = Derivative(Sigma, x)
Piy = Derivative(Sigma, y)
Piz = Derivative(Sigma, z)
Pix = Pix.doit()
Piy = Piy.doit()
Piz = Piz.doit()
Pix = sym.simplify(Pix.expand())
Piy = sym.simplify(Piy.expand())
Piz = sym.simplify(Piz.expand())
self.Pix = Pix
self.Piy = Piy
self.Piz = Piz
self.fPix = lambdify([x, y, z], Pix)
self.fPiy = lambdify([x, y, z], Piy)
self.fPiz = lambdify([x, y, z], Piz)
r*sin(theta)*sin(phi):y,
r*sin(theta)*cos(phi):x})
Sigma=Sigma.subs({r**2*sin(theta)**2:x**2+y**2})
Sigma=Sigma.subs({r**2:x**2+y**2+z**2})
Sigma=sym.simplify(Sigma.expand())
print("Sigma in cartesian coordinates is %s"%Sigma)
Pix=Derivative(Sigma,x)
Piy=Derivative(Sigma,y)
Piz=Derivative(Sigma,z)
Pix=Pix.doit()
Piy=Piy.doit()
Piz=Piz.doit()
Pix=sym.simplify(Pix.expand())
Piy=sym.simplify(Piy.expand())
Piz=sym.simplify(Piz.expand())
print(ell-1,m)
print("Pix is \n%s"%pretty(Pix))
print("Piy is \n%s"%pretty(Piy))
print("Piz is \n%s"%pretty(Piz))
fPix=lambdify([x,y,z],Pix)
fPiy=lambdify([x,y,z],Piy)
fPiz=lambdify([x,y,z],Piz)
xp=1.0
yp=0.0
zp=1.0
