def as_real_imag(self, deep=True, **hints):
# TODO: Handle deep and hints
n, m, theta, phi = self.args
re = (sqrt((2*n + 1)/(4*pi) * C.factorial(n - m)/C.factorial(n + m)) *
C.cos(m*phi) * assoc_legendre(n, m, C.cos(theta)))
im = (sqrt((2*n + 1)/(4*pi) * C.factorial(n - m)/C.factorial(n + m)) *
C.sin(m*phi) * assoc_legendre(n, m, C.cos(theta)))
return (re, im)
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
rv = (sqrt(
(2 * n + 1) / (4 * pi) * factorial(n - m) / factorial(n + m)) *
exp(I * m * phi) * assoc_legendre(n, m, cos(theta)))
# We can do this because of the range of theta
return rv.subs(sqrt(-cos(theta)**2 + 1), sin(theta))
def bp(nu, theta, phi, sigma, l, sigmap, lp, i_n, i, delta):
return (b(l, lp, i_n, i, delta, nu) * legendre(nu, cos(theta)) +
1/(1 + delta**2) * cos(2 * phi) * assoc_legendre(nu, 2, cos(theta)) *
(f(l, l, i_n, i, nu) * (-1)**sigma * kappa(nu, l, l) +
(-1)**(l + lp) * 2 * delta * f(l, lp, i_n, i, nu) *
(-1)**sigmap * kappa(nu, l, lp) +
delta**2 * f(lp, lp, i_n, i, nu) *
(-1)**sigmap * kappa(nu, lp, lp)))
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
rv = (
sqrt((2 * n + 1) / (4 * pi) * C.factorial(n - m) / C.factorial(n + m))
* C.exp(I * m * phi)
* assoc_legendre(n, m, C.cos(theta))
)
# We can do this because of the range of theta
return rv.subs(sqrt(-cos(theta) ** 2 + 1), sin(theta))
def test_special_polynomials():
assert mcode(hermite(x, y)) == "HermiteH[x, y]"
assert mcode(laguerre(x, y)) == "LaguerreL[x, y]"
assert mcode(assoc_laguerre(x, y, z)) == "LaguerreL[x, y, z]"
assert mcode(jacobi(x, y, z, w)) == "JacobiP[x, y, z, w]"
assert mcode(gegenbauer(x, y, z)) == "GegenbauerC[x, y, z]"
assert mcode(chebyshevt(x, y)) == "ChebyshevT[x, y]"
assert mcode(chebyshevu(x, y)) == "ChebyshevU[x, y]"
assert mcode(legendre(x, y)) == "LegendreP[x, y]"
assert mcode(assoc_legendre(x, y, z)) == "LegendreP[x, y, z]"
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
return (sqrt(
(2 * n + 1) / (4 * pi) * C.factorial(n - m) / C.factorial(n + m)) *
C.exp(I * m * phi) * assoc_legendre(n, m, C.cos(theta)))
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
return (sqrt((2*n + 1)/(4*pi) * C.factorial(n - m)/C.factorial(n + m)) *
C.exp(I*m*phi) * assoc_legendre(n, m, C.cos(theta)))