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66 lines
3.8 KiB
66 lines
3.8 KiB
from sympy.core.function import diff
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from sympy.core.numbers import (I, pi)
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from sympy.core.symbol import Symbol
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from sympy.functions.elementary.complexes import conjugate
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import (cos, cot, sin)
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from sympy.functions.special.spherical_harmonics import Ynm, Znm, Ynm_c
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def test_Ynm():
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# https://en.wikipedia.org/wiki/Spherical_harmonics
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th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
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from sympy.abc import n,m
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assert Ynm(0, 0, th, ph).expand(func=True) == 1/(2*sqrt(pi))
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assert Ynm(1, -1, th, ph) == -exp(-2*I*ph)*Ynm(1, 1, th, ph)
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assert Ynm(1, -1, th, ph).expand(func=True) == sqrt(6)*sin(th)*exp(-I*ph)/(4*sqrt(pi))
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assert Ynm(1, 0, th, ph).expand(func=True) == sqrt(3)*cos(th)/(2*sqrt(pi))
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assert Ynm(1, 1, th, ph).expand(func=True) == -sqrt(6)*sin(th)*exp(I*ph)/(4*sqrt(pi))
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assert Ynm(2, 0, th, ph).expand(func=True) == 3*sqrt(5)*cos(th)**2/(4*sqrt(pi)) - sqrt(5)/(4*sqrt(pi))
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assert Ynm(2, 1, th, ph).expand(func=True) == -sqrt(30)*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
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assert Ynm(2, -2, th, ph).expand(func=True) == (-sqrt(30)*exp(-2*I*ph)*cos(th)**2/(8*sqrt(pi))
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+ sqrt(30)*exp(-2*I*ph)/(8*sqrt(pi)))
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assert Ynm(2, 2, th, ph).expand(func=True) == (-sqrt(30)*exp(2*I*ph)*cos(th)**2/(8*sqrt(pi))
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+ sqrt(30)*exp(2*I*ph)/(8*sqrt(pi)))
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assert diff(Ynm(n, m, th, ph), th) == (m*cot(th)*Ynm(n, m, th, ph)
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+ sqrt((-m + n)*(m + n + 1))*exp(-I*ph)*Ynm(n, m + 1, th, ph))
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assert diff(Ynm(n, m, th, ph), ph) == I*m*Ynm(n, m, th, ph)
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assert conjugate(Ynm(n, m, th, ph)) == (-1)**(2*m)*exp(-2*I*m*ph)*Ynm(n, m, th, ph)
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assert Ynm(n, m, -th, ph) == Ynm(n, m, th, ph)
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assert Ynm(n, m, th, -ph) == exp(-2*I*m*ph)*Ynm(n, m, th, ph)
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assert Ynm(n, -m, th, ph) == (-1)**m*exp(-2*I*m*ph)*Ynm(n, m, th, ph)
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def test_Ynm_c():
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th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
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from sympy.abc import n,m
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assert Ynm_c(n, m, th, ph) == (-1)**(2*m)*exp(-2*I*m*ph)*Ynm(n, m, th, ph)
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def test_Znm():
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# https://en.wikipedia.org/wiki/Solid_harmonics#List_of_lowest_functions
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th, ph = Symbol("theta", real=True), Symbol("phi", real=True)
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assert Znm(0, 0, th, ph) == Ynm(0, 0, th, ph)
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assert Znm(1, -1, th, ph) == (-sqrt(2)*I*(Ynm(1, 1, th, ph)
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- exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
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assert Znm(1, 0, th, ph) == Ynm(1, 0, th, ph)
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assert Znm(1, 1, th, ph) == (sqrt(2)*(Ynm(1, 1, th, ph)
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+ exp(-2*I*ph)*Ynm(1, 1, th, ph))/2)
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assert Znm(0, 0, th, ph).expand(func=True) == 1/(2*sqrt(pi))
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assert Znm(1, -1, th, ph).expand(func=True) == (sqrt(3)*I*sin(th)*exp(I*ph)/(4*sqrt(pi))
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- sqrt(3)*I*sin(th)*exp(-I*ph)/(4*sqrt(pi)))
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assert Znm(1, 0, th, ph).expand(func=True) == sqrt(3)*cos(th)/(2*sqrt(pi))
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assert Znm(1, 1, th, ph).expand(func=True) == (-sqrt(3)*sin(th)*exp(I*ph)/(4*sqrt(pi))
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- sqrt(3)*sin(th)*exp(-I*ph)/(4*sqrt(pi)))
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assert Znm(2, -1, th, ph).expand(func=True) == (sqrt(15)*I*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
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- sqrt(15)*I*sin(th)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))
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assert Znm(2, 0, th, ph).expand(func=True) == 3*sqrt(5)*cos(th)**2/(4*sqrt(pi)) - sqrt(5)/(4*sqrt(pi))
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assert Znm(2, 1, th, ph).expand(func=True) == (-sqrt(15)*sin(th)*exp(I*ph)*cos(th)/(4*sqrt(pi))
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- sqrt(15)*sin(th)*exp(-I*ph)*cos(th)/(4*sqrt(pi)))
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