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| 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 | |
| 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) | |