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_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_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_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_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)
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 _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