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from sympy.core.evalf import Nfrom sympy.core.function import (Derivative, Function, PoleError, Subs)from sympy.core.numbers import (E, Rational, oo, pi)from sympy.core.singleton import Sfrom sympy.core.symbol import (Symbol, symbols)from sympy.functions.elementary.exponential import (LambertW, exp, log)from sympy.functions.elementary.miscellaneous import sqrtfrom sympy.functions.elementary.trigonometric import (atan, cos, sin)from sympy.integrals.integrals import Integralfrom sympy.series.order import Ofrom sympy.series.series import seriesfrom sympy.abc import x, y, n, kfrom sympy.testing.pytest import raisesfrom 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.Zero.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
raises(PoleError, lambda: ((1/sin(x))**oo).series()) 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).removeO()
def test_series_of_Subs(): from sympy.abc import 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 subs1_series = (Subs(x, x, y) + Subs(-x**3/6, x, y) + Subs(x**5/120, x, y) + O(y**6)) assert subs1.series() == subs1_series assert subs1.series(y) == subs1_series 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)
raises(ValueError, lambda: Subs(x + 2*y, y, z).series()) assert Subs(x + y, y, z).series(x).doit() == x + z
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, shanksfrom sympy.concrete.summations import Sumfrom sympy.core.numbers import 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)**Rational(2, 3) assert (eq + 1).as_leading_term(x) == eq
def test_x_is_base_detection(): eq = (x**2)**Rational(2, 3) assert eq.series() == x**Rational(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_9549(): y = (x**2 + x + 1) / (x**3 + x**2) assert series(y, x, oo) == x**(-5) - 1/x**4 + x**(-3) + 1/x + O(x**(-6), (x, oo))
def test_issue_10761(): assert series(1/(x**-2 + x**-3), x, 0) == x**3 - x**4 + x**5 + O(x**6)
def test_issue_12578(): y = (1 - 1/(x/2 - 1/(2*x))**4)**(S(1)/8) assert y.series(x, 0, n=17) == 1 - 2*x**4 - 8*x**6 - 34*x**8 - 152*x**10 - 714*x**12 - \ 3472*x**14 - 17318*x**16 + O(x**17)
def test_issue_12791(): beta = symbols('beta', positive=True) theta, varphi = symbols('theta varphi', real=True)
expr = (-beta**2*varphi*sin(theta) + beta**2*cos(theta) + \ beta*varphi*sin(theta) - beta*cos(theta) - beta + 1)/(beta*cos(theta) - 1)**2
sol = 0.5/(0.5*cos(theta) - 1.0)**2 - 0.25*cos(theta)/(0.5*cos(theta)\ - 1.0)**2 + (beta - 0.5)*(-0.25*varphi*sin(2*theta) - 1.5*cos(theta)\ + 0.25*cos(2*theta) + 1.25)/(0.5*cos(theta) - 1.0)**3\ + 0.25*varphi*sin(theta)/(0.5*cos(theta) - 1.0)**2 + O((beta - S.Half)**2, (beta, S.Half))
assert expr.series(beta, 0.5, 2).trigsimp() == sol
def test_issue_14885(): assert series(x**Rational(-3, 2)*exp(x), x, 0) == (x**Rational(-3, 2) + 1/sqrt(x) + sqrt(x)/2 + x**Rational(3, 2)/6 + x**Rational(5, 2)/24 + x**Rational(7, 2)/120 + x**Rational(9, 2)/720 + x**Rational(11, 2)/5040 + O(x**6))
def test_issue_15539(): assert series(atan(x), x, -oo) == (-1/(5*x**5) + 1/(3*x**3) - 1/x - pi/2 + O(x**(-6), (x, -oo))) assert series(atan(x), x, oo) == (-1/(5*x**5) + 1/(3*x**3) - 1/x + pi/2 + O(x**(-6), (x, oo)))
def test_issue_7259(): assert series(LambertW(x), x) == x - x**2 + 3*x**3/2 - 8*x**4/3 + 125*x**5/24 + O(x**6) assert series(LambertW(x**2), x, n=8) == x**2 - x**4 + 3*x**6/2 + O(x**8) assert series(LambertW(sin(x)), x, n=4) == x - x**2 + 4*x**3/3 + O(x**4)
def test_issue_11884(): assert cos(x).series(x, 1, n=1) == cos(1) + O(x - 1, (x, 1))
def test_issue_18008(): y = x*(1 + x*(1 - x))/((1 + x*(1 - x)) - (1 - x)*(1 - x)) assert y.series(x, oo, n=4) == -9/(32*x**3) - 3/(16*x**2) - 1/(8*x) + S(1)/4 + x/2 + \ O(x**(-4), (x, oo))
def test_issue_18842(): f = log(x/(1 - x)) assert f.series(x, 0.491, n=1).removeO().nsimplify() == \ -S(180019443780011)/5000000000000000
def test_issue_19534(): dt = symbols('dt', real=True) expr = 16*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0)/45 + \ 49*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.96296296296296296296*dt + 1.0) + 0.051640768506639183825*dt + \ dt*(1/2 - sqrt(21)/14) + 1.0)/180 + 49*dt*(-0.23637909581542530626*dt*(2.0*dt + 1.0) - \ 0.74817562366625959291*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.88085458023927036857*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.96296296296296296296*dt + 1.0) + \ 2.1165151389911680013*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.96296296296296296296*dt + 1.0) + 0.22431393315265061193*dt + 1.0) - \ 1.1854881643947648988*dt + dt*(sqrt(21)/14 + 1/2) + 1.0)/180 + \ dt*(0.66666666666666666667*dt*(2.0*dt + 1.0) + \ 6.0173399699313066769*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ 4.1117044797036320069*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.96296296296296296296*dt + 1.0) - \ 7.0189140975801991157*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.96296296296296296296*dt + 1.0) + 0.22431393315265061193*dt + 1.0) + \ 0.94010945196161777522*dt*(-0.23637909581542530626*dt*(2.0*dt + 1.0) - \ 0.74817562366625959291*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.88085458023927036857*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.96296296296296296296*dt + 1.0) + \ 2.1165151389911680013*dt*(-0.049335189898860408029*dt*(2.0*dt + 1.0) + \ 0.29601113939316244817*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) - \ 0.12564355335492979587*dt*(0.074074074074074074074*dt*(2.0*dt + 1.0) + \ 0.2962962962962962963*dt*(0.125*dt*(2.0*dt + 1.0) + 0.875*dt + 1.0) + \ 0.96296296296296296296*dt + 1.0) + 0.22431393315265061193*dt + 1.0) - \ 0.35816132904077632692*dt + 1.0) + 5.5065024887242400038*dt + 1.0)/20 + dt/20 + 1
assert N(expr.series(dt, 0, 8), 20) == -0.00092592592592592596126*dt**7 + 0.0027777777777777783175*dt**6 + \ 0.016666666666666656027*dt**5 + 0.083333333333333300952*dt**4 + 0.33333333333333337034*dt**3 + \ 1.0*dt**2 + 1.0*dt + 1.0
def test_issue_11407(): a, b, c, x = symbols('a b c x') assert series(sqrt(a + b + c*x), x, 0, 1) == sqrt(a + b) + O(x) assert series(sqrt(a + b + c + c*x), x, 0, 1) == sqrt(a + b + c) + O(x)
def test_issue_14037(): assert (sin(x**50)/x**51).series(x, n=0) == 1/x + O(1, x)
def test_issue_20551(): expr = (exp(x)/x).series(x, n=None) terms = [ next(expr) for i in range(3) ] assert terms == [1/x, 1, x/2]
def test_issue_20697(): p_0, p_1, p_2, p_3, b_0, b_1, b_2 = symbols('p_0 p_1 p_2 p_3 b_0 b_1 b_2') Q = (p_0 + (p_1 + (p_2 + p_3/y)/y)/y)/(1 + ((p_3/(b_0*y) + (b_0*p_2\ - b_1*p_3)/b_0**2)/y + (b_0**2*p_1 - b_0*b_1*p_2 - p_3*(b_0*b_2\ - b_1**2))/b_0**3)/y) assert Q.series(y, n=3).ratsimp() == b_2*y**2 + b_1*y + b_0 + O(y**3)
def test_issue_21245(): fi = (1 + sqrt(5))/2 assert (1/(1 - x - x**2)).series(x, 1/fi, 1).factor() == \ (-4812 - 2152*sqrt(5) + 1686*x + 754*sqrt(5)*x\ + O((x - 2/(1 + sqrt(5)))**2, (x, 2/(1 + sqrt(5)))))/((1 + sqrt(5))\ *(20 + 9*sqrt(5))**2*(x + sqrt(5)*x - 2))
def test_issue_21938(): expr = sin(1/x + exp(-x)) - sin(1/x) assert expr.series(x, oo) == (1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x)
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