from sympy import sin, cos, exp, E, series, oo, S, Derivative, O, Integral, \ Function, log, sqrt, Symbol, Subs, pi, symbols, IndexedBase from sympy.abc import x, y, n, k from sympy.utilities.pytest import raises from sympy.core.compatibility import range from sympy.series.gruntz import calculate_series def test_sin(): e1 = sin(x).series(x, 0) e2 = series(sin(x), x, 0) assert e1 == e2 def test_cos(): e1 = cos(x).series(x, 0) e2 = series(cos(x), x, 0) assert e1 == e2 def test_exp(): e1 = exp(x).series(x, 0) e2 = series(exp(x), x, 0) assert e1 == e2 def test_exp2(): e1 = exp(cos(x)).series(x, 0) e2 = series(exp(cos(x)), x, 0) assert e1 == e2 def test_issue_5223(): assert series(1, x) == 1 assert next(S(0).lseries(x)) == 0 assert cos(x).series() == cos(x).series(x) raises(ValueError, lambda: cos(x + y).series()) raises(ValueError, lambda: x.series(dir="")) assert (cos(x).series(x, 1) - cos(x + 1).series(x).subs(x, x - 1)).removeO() == 0 e = cos(x).series(x, 1, n=None) assert [next(e) for i in range(2)] == [cos(1), -((x - 1)*sin(1))] e = cos(x).series(x, 1, n=None, dir='-') assert [next(e) for i in range(2)] == [cos(1), (1 - x)*sin(1)] # the following test is exact so no need for x -> x - 1 replacement assert abs(x).series(x, 1, dir='-') == x assert exp(x).series(x, 1, dir='-', n=3).removeO() == \ E - E*(-x + 1) + E*(-x + 1)**2/2 D = Derivative assert D(x**2 + x**3*y**2, x, 2, y, 1).series(x).doit() == 12*x*y assert next(D(cos(x), x).lseries()) == D(1, x) assert D( exp(x), x).series(n=3) == D(1, x) + D(x, x) + D(x**2/2, x) + D(x**3/6, x) + O(x**3) assert Integral(x, (x, 1, 3), (y, 1, x)).series(x) == -4 + 4*x assert (1 + x + O(x**2)).getn() == 2 assert (1 + x).getn() is None assert ((1/sin(x))**oo).series() == oo logx = Symbol('logx') assert ((sin(x))**y).nseries(x, n=1, logx=logx) == \ exp(y*logx) + O(x*exp(y*logx), x) assert sin(1/x).series(x, oo, n=5) == 1/x - 1/(6*x**3) + O(x**(-5), (x, oo)) assert abs(x).series(x, oo, n=5, dir='+') == x assert abs(x).series(x, -oo, n=5, dir='-') == -x assert abs(-x).series(x, oo, n=5, dir='+') == x assert abs(-x).series(x, -oo, n=5, dir='-') == -x assert exp(x*log(x)).series(n=3) == \ 1 + x*log(x) + x**2*log(x)**2/2 + O(x**3*log(x)**3) # XXX is this right? If not, fix "ngot > n" handling in expr. p = Symbol('p', positive=True) assert exp(sqrt(p)**3*log(p)).series(n=3) == \ 1 + p**S('3/2')*log(p) + O(p**3*log(p)**3) assert exp(sin(x)*log(x)).series(n=2) == 1 + x*log(x) + O(x**2*log(x)**2) def test_issue_11313(): assert Integral(cos(x), x).series(x) == sin(x).series(x) assert Derivative(sin(x), x).series(x, n=3).doit() == cos(x).series(x, n=3) assert Derivative(x**3, x).as_leading_term(x) == 3*x**2 assert Derivative(x**3, y).as_leading_term(x) == 0 assert Derivative(sin(x), x).as_leading_term(x) == 1 assert Derivative(cos(x), x).as_leading_term(x) == -x # This result is equivalent to zero, zero is not return because # `Expr.series` doesn't currently detect an `x` in its `free_symbol`s. assert Derivative(1, x).as_leading_term(x) == Derivative(1, x) assert Derivative(exp(x), x).series(x).doit() == exp(x).series(x) assert 1 + Integral(exp(x), x).series(x) == exp(x).series(x) assert Derivative(log(x), x).series(x).doit() == (1/x).series(x) assert Integral(log(x), x).series(x) == Integral(log(x), x).doit().series(x) def test_series_of_Subs(): from sympy.abc import x, y, z subs1 = Subs(sin(x), (x,), (y,)) subs2 = Subs(sin(x) * cos(z), (x,), (y,)) subs3 = Subs(sin(x * z), (x, z), (y, x)) assert subs1.series(x) == subs1 assert subs1.series(y) == Subs(x, (x,), (y,)) + Subs(-x**3/6, (x,), (y,)) + Subs(x**5/120, (x,), (y,)) + O(y**6) assert subs1.series(z) == subs1 assert subs2.series(z) == Subs(z**4*sin(x)/24, (x,), (y,)) + Subs(-z**2*sin(x)/2, (x,), (y,)) + Subs(sin(x), (x,), (y,)) + O(z**6) assert subs3.series(x).doit() == subs3.doit().series(x) assert subs3.series(z).doit() == sin(x*y) def test_issue_3978(): f = Function('f') assert f(x).series(x, 0, 3, dir='-') == \ f(0) + x*Subs(Derivative(f(x), x), (x,), (0,)) + \ x**2*Subs(Derivative(f(x), x, x), (x,), (0,))/2 + O(x**3) assert f(x).series(x, 0, 3) == \ f(0) + x*Subs(Derivative(f(x), x), (x,), (0,)) + \ x**2*Subs(Derivative(f(x), x, x), (x,), (0,))/2 + O(x**3) assert f(x**2).series(x, 0, 3) == \ f(0) + x**2*Subs(Derivative(f(x), x), (x,), (0,)) + O(x**3) assert f(x**2+1).series(x, 0, 3) == \ f(1) + x**2*Subs(Derivative(f(x), x), (x,), (1,)) + O(x**3) class TestF(Function): pass assert TestF(x).series(x, 0, 3) == TestF(0) + \ x*Subs(Derivative(TestF(x), x), (x,), (0,)) + \ x**2*Subs(Derivative(TestF(x), x, x), (x,), (0,))/2 + O(x**3) from sympy.series.acceleration import richardson, shanks from sympy import Sum, Integer def test_acceleration(): e = (1 + 1/n)**n assert round(richardson(e, n, 10, 20).evalf(), 10) == round(E.evalf(), 10) A = Sum(Integer(-1)**(k + 1) / k, (k, 1, n)) assert round(shanks(A, n, 25).evalf(), 4) == round(log(2).evalf(), 4) assert round(shanks(A, n, 25, 5).evalf(), 10) == round(log(2).evalf(), 10) def test_issue_5852(): assert series(1/cos(x/log(x)), x, 0) == 1 + x**2/(2*log(x)**2) + \ 5*x**4/(24*log(x)**4) + O(x**6) def test_issue_4583(): assert cos(1 + x + x**2).series(x, 0, 5) == cos(1) - x*sin(1) + \ x**2*(-sin(1) - cos(1)/2) + x**3*(-cos(1) + sin(1)/6) + \ x**4*(-11*cos(1)/24 + sin(1)/2) + O(x**5) def test_issue_6318(): eq = (1/x)**(S(2)/3) assert (eq + 1).as_leading_term(x) == eq def test_x_is_base_detection(): eq = (x**2)**(S(2)/3) assert eq.series() == x**(S(4)/3) def test_sin_power(): e = sin(x)**1.2 assert calculate_series(e, x) == x**1.2 def test_issue_7203(): assert series(cos(x), x, pi, 3) == \ -1 + (x - pi)**2/2 + O((x - pi)**3, (x, pi)) def test_exp_product_positive_factors(): a, b = symbols('a, b', positive=True) x = a * b assert series(exp(x), x, n=8) == 1 + a*b + a**2*b**2/2 + \ a**3*b**3/6 + a**4*b**4/24 + a**5*b**5/120 + a**6*b**6/720 + \ a**7*b**7/5040 + O(a**8*b**8, a, b) def test_issue_8805(): assert series(1, n=8) == 1 def test_issue_10761(): assert series(1/(x**-2 + x**-3), x, 0) == x**3 - x**4 + x**5 + O(x**6)