from sympy.core.add import Add from sympy.core.numbers import (Rational, oo, pi) from sympy.core.singleton import S from sympy.core.symbol import symbols from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.piecewise import Piecewise from sympy.functions.elementary.trigonometric import (cos, sin, sinc, tan) from sympy.series.fourier import fourier_series from sympy.series.fourier import FourierSeries from sympy.testing.pytest import raises from functools import lru_cache x, y, z = symbols('x y z') # Don't declare these during import because they are slow @lru_cache() def _get_examples(): fo = fourier_series(x, (x, -pi, pi)) fe = fourier_series(x**2, (-pi, pi)) fp = fourier_series(Piecewise((0, x < 0), (pi, True)), (x, -pi, pi)) return fo, fe, fp def test_FourierSeries(): fo, fe, fp = _get_examples() assert fourier_series(1, (-pi, pi)) == 1 assert (Piecewise((0, x < 0), (pi, True)). fourier_series((x, -pi, pi)).truncate()) == fp.truncate() assert isinstance(fo, FourierSeries) assert fo.function == x assert fo.x == x assert fo.period == (-pi, pi) assert fo.term(3) == 2*sin(3*x) / 3 assert fe.term(3) == -4*cos(3*x) / 9 assert fp.term(3) == 2*sin(3*x) / 3 assert fo.as_leading_term(x) == 2*sin(x) assert fe.as_leading_term(x) == pi**2 / 3 assert fp.as_leading_term(x) == pi / 2 assert fo.truncate() == 2*sin(x) - sin(2*x) + (2*sin(3*x) / 3) assert fe.truncate() == -4*cos(x) + cos(2*x) + pi**2 / 3 assert fp.truncate() == 2*sin(x) + (2*sin(3*x) / 3) + pi / 2 fot = fo.truncate(n=None) s = [0, 2*sin(x), -sin(2*x)] for i, t in enumerate(fot): if i == 3: break assert s[i] == t def _check_iter(f, i): for ind, t in enumerate(f): assert t == f[ind] if ind == i: break _check_iter(fo, 3) _check_iter(fe, 3) _check_iter(fp, 3) assert fo.subs(x, x**2) == fo raises(ValueError, lambda: fourier_series(x, (0, 1, 2))) raises(ValueError, lambda: fourier_series(x, (x, 0, oo))) raises(ValueError, lambda: fourier_series(x*y, (0, oo))) def test_FourierSeries_2(): p = Piecewise((0, x < 0), (x, True)) f = fourier_series(p, (x, -2, 2)) assert f.term(3) == (2*sin(3*pi*x / 2) / (3*pi) - 4*cos(3*pi*x / 2) / (9*pi**2)) assert f.truncate() == (2*sin(pi*x / 2) / pi - sin(pi*x) / pi - 4*cos(pi*x / 2) / pi**2 + S.Half) def test_square_wave(): """Test if fourier_series approximates discontinuous function correctly.""" square_wave = Piecewise((1, x < pi), (-1, True)) s = fourier_series(square_wave, (x, 0, 2*pi)) assert s.truncate(3) == 4 / pi * sin(x) + 4 / (3 * pi) * sin(3 * x) + \ 4 / (5 * pi) * sin(5 * x) assert s.sigma_approximation(4) == 4 / pi * sin(x) * sinc(pi / 4) + \ 4 / (3 * pi) * sin(3 * x) * sinc(3 * pi / 4) def test_sawtooth_wave(): s = fourier_series(x, (x, 0, pi)) assert s.truncate(4) == \ pi/2 - sin(2*x) - sin(4*x)/2 - sin(6*x)/3 s = fourier_series(x, (x, 0, 1)) assert s.truncate(4) == \ S.Half - sin(2*pi*x)/pi - sin(4*pi*x)/(2*pi) - sin(6*pi*x)/(3*pi) def test_FourierSeries__operations(): fo, fe, fp = _get_examples() fes = fe.scale(-1).shift(pi**2) assert fes.truncate() == 4*cos(x) - cos(2*x) + 2*pi**2 / 3 assert fp.shift(-pi/2).truncate() == (2*sin(x) + (2*sin(3*x) / 3) + (2*sin(5*x) / 5)) fos = fo.scale(3) assert fos.truncate() == 6*sin(x) - 3*sin(2*x) + 2*sin(3*x) fx = fe.scalex(2).shiftx(1) assert fx.truncate() == -4*cos(2*x + 2) + cos(4*x + 4) + pi**2 / 3 fl = fe.scalex(3).shift(-pi).scalex(2).shiftx(1).scale(4) assert fl.truncate() == (-16*cos(6*x + 6) + 4*cos(12*x + 12) - 4*pi + 4*pi**2 / 3) raises(ValueError, lambda: fo.shift(x)) raises(ValueError, lambda: fo.shiftx(sin(x))) raises(ValueError, lambda: fo.scale(x*y)) raises(ValueError, lambda: fo.scalex(x**2)) def test_FourierSeries__neg(): fo, fe, fp = _get_examples() assert (-fo).truncate() == -2*sin(x) + sin(2*x) - (2*sin(3*x) / 3) assert (-fe).truncate() == +4*cos(x) - cos(2*x) - pi**2 / 3 def test_FourierSeries__add__sub(): fo, fe, fp = _get_examples() assert fo + fo == fo.scale(2) assert fo - fo == 0 assert -fe - fe == fe.scale(-2) assert (fo + fe).truncate() == 2*sin(x) - sin(2*x) - 4*cos(x) + cos(2*x) \ + pi**2 / 3 assert (fo - fe).truncate() == 2*sin(x) - sin(2*x) + 4*cos(x) - cos(2*x) \ - pi**2 / 3 assert isinstance(fo + 1, Add) raises(ValueError, lambda: fo + fourier_series(x, (x, 0, 2))) def test_FourierSeries_finite(): assert fourier_series(sin(x)).truncate(1) == sin(x) # assert type(fourier_series(sin(x)*log(x))).truncate() == FourierSeries # assert type(fourier_series(sin(x**2+6))).truncate() == FourierSeries assert fourier_series(sin(x)*log(y)*exp(z),(x,pi,-pi)).truncate() == sin(x)*log(y)*exp(z) assert fourier_series(sin(x)**6).truncate(oo) == -15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 \ + Rational(5, 16) assert fourier_series(sin(x) ** 6).truncate() == -15 * cos(2 * x) / 32 + 3 * cos(4 * x) / 16 \ + Rational(5, 16) assert fourier_series(sin(4*x+3) + cos(3*x+4)).truncate(oo) == -sin(4)*sin(3*x) + sin(4*x)*cos(3) \ + cos(4)*cos(3*x) + sin(3)*cos(4*x) assert fourier_series(sin(x)+cos(x)*tan(x)).truncate(oo) == 2*sin(x) assert fourier_series(cos(pi*x), (x, -1, 1)).truncate(oo) == cos(pi*x) assert fourier_series(cos(3*pi*x + 4) - sin(4*pi*x)*log(pi*y), (x, -1, 1)).truncate(oo) == -log(pi*y)*sin(4*pi*x)\ - sin(4)*sin(3*pi*x) + cos(4)*cos(3*pi*x)