phivenv/Lib/site-packages/sympy/functions/special/tests/test_spherical_harmonics.py

from sympy.core.function import diff
from sympy.core.numbers import (I, pi)
from sympy.core.symbol import Symbol
from sympy.functions.elementary.complexes import conjugate
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, cot, sin)
from sympy.functions.special.spherical_harmonics import Ynm, Znm, Ynm_c


def test_Ynm():
    # https://en.wikipedia.org/wiki/Spherical_harmonics
    th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
    from sympy.abc import n,m

    assert Ynm(0, 0, th, ph).expand(func=True) == 1/(2*sqrt(pi))
    assert Ynm(1, -1, th, ph) == -exp(-2*I*ph)*Ynm(1, 1, th, ph)
    assert Ynm(1, -1, th, ph).expand(func=True) == sqrt(6)*sin(th)*exp(-I*ph)/(4*sqrt(pi))
    assert Ynm(1, 0, th, ph).expand(func=True) == sqrt(3)*cos(th)/(2*sqrt(pi))
    assert Ynm(1, 1, th, ph).expand(func=True) == -sqrt(6)*sin(th)*exp(I*ph)/(4*sqrt(pi))
    assert Ynm(2, 0, th, ph).expand(func=True) == 3*sqrt(5)*cos(th)**2/(4*sqrt(pi)) - sqrt(5)/(4*sqrt(pi))
    assert Ynm(2, 1, th, ph).expand(func=True) == -sqrt(30)*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
    assert Ynm(2, -2, th, ph).expand(func=True) == (-sqrt(30)*exp(-2*I*ph)*cos(th)**2/(8*sqrt(pi))
                                                    + sqrt(30)*exp(-2*I*ph)/(8*sqrt(pi)))
    assert Ynm(2, 2, th, ph).expand(func=True) == (-sqrt(30)*exp(2*I*ph)*cos(th)**2/(8*sqrt(pi))
                                                   + sqrt(30)*exp(2*I*ph)/(8*sqrt(pi)))

    assert diff(Ynm(n, m, th, ph), th) == (m*cot(th)*Ynm(n, m, th, ph)
                                           + sqrt((-m + n)*(m + n + 1))*exp(-I*ph)*Ynm(n, m + 1, th, ph))
    assert diff(Ynm(n, m, th, ph), ph) == I*m*Ynm(n, m, th, ph)

    assert conjugate(Ynm(n, m, th, ph)) == (-1)**(2*m)*exp(-2*I*m*ph)*Ynm(n, m, th, ph)

    assert Ynm(n, m, -th, ph) == Ynm(n, m, th, ph)
    assert Ynm(n, m, th, -ph) == exp(-2*I*m*ph)*Ynm(n, m, th, ph)
    assert Ynm(n, -m, th, ph) == (-1)**m*exp(-2*I*m*ph)*Ynm(n, m, th, ph)


def test_Ynm_c():
    th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
    from sympy.abc import n,m

    assert Ynm_c(n, m, th, ph) == (-1)**(2*m)*exp(-2*I*m*ph)*Ynm(n, m, th, ph)


def test_Znm():
    # https://en.wikipedia.org/wiki/Solid_harmonics#List_of_lowest_functions
    th, ph = Symbol("theta", real=True), Symbol("phi", real=True)

    assert Znm(0, 0, th, ph) == Ynm(0, 0, th, ph)
    assert Znm(1, -1, th, ph) == (-sqrt(2)*I*(Ynm(1, 1, th, ph)
                                  - exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
    assert Znm(1, 0, th, ph) == Ynm(1, 0, th, ph)
    assert Znm(1, 1, th, ph) == (sqrt(2)*(Ynm(1, 1, th, ph)
                                 + exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
    assert Znm(0, 0, th, ph).expand(func=True) == 1/(2*sqrt(pi))
    assert Znm(1, -1, th, ph).expand(func=True) == (sqrt(3)*I*sin(th)*exp(I*ph)/(4*sqrt(pi))
                                                    - sqrt(3)*I*sin(th)*exp(-I*ph)/(4*sqrt(pi)))
    assert Znm(1, 0, th, ph).expand(func=True) == sqrt(3)*cos(th)/(2*sqrt(pi))
    assert Znm(1, 1, th, ph).expand(func=True) == (-sqrt(3)*sin(th)*exp(I*ph)/(4*sqrt(pi))
                                                   - sqrt(3)*sin(th)*exp(-I*ph)/(4*sqrt(pi)))
    assert Znm(2, -1, th, ph).expand(func=True) == (sqrt(15)*I*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
                                                    - sqrt(15)*I*sin(th)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))
    assert Znm(2, 0, th, ph).expand(func=True) == 3*sqrt(5)*cos(th)**2/(4*sqrt(pi)) - sqrt(5)/(4*sqrt(pi))
    assert Znm(2, 1, th, ph).expand(func=True) == (-sqrt(15)*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
                                                   - sqrt(15)*sin(th)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))