def _eval_(self, n, m, theta, phi, **kwargs):
r"""
TESTS::
sage: x, y = var('x y')
sage: spherical_harmonic(1, 2, x, y)
0
sage: spherical_harmonic(1, -2, x, y)
0
sage: spherical_harmonic(1/2, 2, x, y)
spherical_harmonic(1/2, 2, x, y)
sage: spherical_harmonic(3, 2, x, y)
1/8*sqrt(30)*sqrt(7)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
sage: spherical_harmonic(3, 2, 1, 2)
1/8*sqrt(30)*sqrt(7)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
sage: spherical_harmonic(3 + I, 2., 1, 2)
-0.351154337307488 - 0.415562233975369*I
Check that :trac:`20939` is fixed::
sage: ex = spherical_harmonic(3,2,1,2*pi/3)
sage: QQbar(ex * sqrt(pi)/cos(1)/sin(1)^2).minpoly()
x^4 + 105/32*x^2 + 11025/1024
"""
if n in ZZ and m in ZZ and n > -1:
if abs(m) > n:
return ZZ(0)
if m == 0 and theta.is_zero():
return sqrt((2*n+1)/4/pi)
from sage.arith.misc import factorial
from sage.functions.trig import cos
from sage.functions.orthogonal_polys import gen_legendre_P
return (sqrt(factorial(n-m) * (2*n+1) / (4*pi * factorial(n+m))) *
exp(I*m*phi) * gen_legendre_P(n, m, cos(theta)) *
(-1)**m).simplify_trig()
def _eval_(self, n, m, theta, phi, **kwargs):
r"""
TESTS::
sage: x, y = var('x y')
sage: spherical_harmonic(1, 2, x, y)
0
sage: spherical_harmonic(1, -2, x, y)
0
sage: spherical_harmonic(1/2, 2, x, y)
spherical_harmonic(1/2, 2, x, y)
sage: spherical_harmonic(3, 2, x, y)
1/8*sqrt(30)*sqrt(7)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
sage: spherical_harmonic(3, 2, 1, 2)
1/8*sqrt(30)*sqrt(7)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
sage: spherical_harmonic(3 + I, 2., 1, 2)
-0.351154337307488 - 0.415562233975369*I
Check that :trac:`20939` is fixed::
sage: ex = spherical_harmonic(3,2,1,2*pi/3)
sage: QQbar(ex * sqrt(pi)/cos(1)/sin(1)^2).minpoly()
x^4 + 105/32*x^2 + 11025/1024
"""
if n in ZZ and m in ZZ and n > -1:
if abs(m) > n:
return ZZ(0)
if m == 0 and theta.is_zero():
return sqrt((2*n+1)/4/pi)
from sage.arith.misc import factorial
from sage.functions.trig import cos
from sage.functions.orthogonal_polys import gen_legendre_P
return (sqrt(factorial(n-m) * (2*n+1) / (4*pi * factorial(n+m))) *
exp(I*m*phi) * gen_legendre_P(n, m, cos(theta)) *
(-1)**m).simplify_trig()
