def test_simple(): # issue sympy/sympy#5223 assert series(1, x) == 1 pytest.raises(ValueError, lambda: cos(x + y).series()) pytest.raises(ValueError, lambda: x.series(dir='')) D = Derivative assert D(x**2 + x**3*y**2, x, 2, y, 1).series(x).doit().removeO() == 12*x*y assert (1 + x).getn() is None # issue sympy/sympy#8805 assert series(1, n=8) == 1 # issue sympy/sympy#5223 assert (cos(x).series(x, 1) - cos(x + 1).series(x).subs({x: x - 1})).removeO() == 0 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 # issue sympy/sympy#7203 assert series(cos(x), x, pi, 3) == -1 + (x - pi)**2/2 + O((x - pi)**3, (x, pi))
def test_sin(): e1 = sin(x).series(x, 0) e2 = series(sin(x), x, 0) assert e1 == e2 # issue sympy/sympy#5223: assert ((1/sin(x))**oo).series() == oo
def test_airybi(): z = Symbol('z', extended_real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybi(z), airybi) assert airybi(0) == 3**Rational(5, 6) / (3 * gamma(Rational(2, 3))) assert airybi(oo) == oo assert airybi(-oo) == 0 assert diff(airybi(z), z) == airybiprime(z) assert series(airybi(z), z, 0, 3) == (cbrt(3) * gamma(Rational(1, 3)) / (2 * pi) + 3**Rational(2, 3) * z * gamma(Rational(2, 3)) / (2 * pi) + O(z**3)) l = Limit( airybi(I / x) / (exp(Rational(2, 3) * (I / x)**Rational(3, 2)) * sqrt(pi * sqrt(I / x))), x, 0) assert l.doit() == l assert airybi(z).rewrite(hyper) == (root(3, 6) * z * hyper( (), (Rational(4, 3), ), z**3 / 9) / gamma(Rational(1, 3)) + 3**Rational(5, 6) * hyper( (), (Rational(2, 3), ), z**3 / 9) / (3 * gamma(Rational(2, 3)))) assert isinstance(airybi(z).rewrite(besselj), airybi) assert (airybi(t).rewrite(besselj) == sqrt(3) * sqrt(-t) * (besselj(-1 / 3, 2 * (-t)**Rational(3, 2) / 3) - besselj(Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3) assert airybi(z).rewrite(besseli) == ( sqrt(3) * (z * besseli(Rational(1, 3), 2 * z**Rational(3, 2) / 3) / cbrt(z**Rational(3, 2)) + cbrt(z**Rational(3, 2)) * besseli(-Rational(1, 3), 2 * z**Rational(3, 2) / 3)) / 3) assert airybi(p).rewrite(besseli) == ( sqrt(3) * sqrt(p) * (besseli(-Rational(1, 3), 2 * p**Rational(3, 2) / 3) + besseli(Rational(1, 3), 2 * p**Rational(3, 2) / 3)) / 3) assert airybi(p).rewrite(besselj) == airybi(p) assert expand_func(airybi( 2 * cbrt(3 * z**5))) == (sqrt(3) * (1 - cbrt(z**5) / z**Rational(5, 3)) * airyai(2 * cbrt(3) * z**Rational(5, 3)) / 2 + (1 + cbrt(z**5) / z**Rational(5, 3)) * airybi(2 * cbrt(3) * z**Rational(5, 3)) / 2) assert expand_func(airybi(x * y)) == airybi(x * y) assert expand_func(airybi(log(x))) == airybi(log(x)) assert expand_func(airybi(2 * root(3 * z**5, 5))) == airybi( 2 * root(3 * z**5, 5)) assert airybi(x).taylor_term(-1, x) == 0
def test_airyai(): z = Symbol('z', extended_real=False) r = Symbol('r', extended_real=True) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airyai(z), airyai) assert airyai(0) == cbrt(3)/(3*gamma(Rational(2, 3))) assert airyai(oo) == 0 assert airyai(-oo) == 0 assert diff(airyai(z), z) == airyaiprime(z) assert series(airyai(z), z, 0, 3) == ( 3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) - root(3, 6)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3)) l = Limit(airyai(I/x)/(exp(-Rational(2, 3)*(I/x)**Rational(3, 2))*sqrt(pi*sqrt(I/x))/2), x, 0) assert l.doit() == l # cover _airyais._eval_aseries assert airyai(z).rewrite(hyper) == ( -3**Rational(2, 3)*z*hyper((), (Rational(4, 3),), z**3/9)/(3*gamma(Rational(1, 3))) + cbrt(3)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3)))) assert isinstance(airyai(z).rewrite(besselj), airyai) assert airyai(t).rewrite(besselj) == ( sqrt(-t)*(besselj(-Rational(1, 3), 2*(-t)**Rational(3, 2)/3) + besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3) assert airyai(z).rewrite(besseli) == ( -z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/(3*cbrt(z**Rational(3, 2))) + cbrt(z**Rational(3, 2))*besseli(-Rational(1, 3), 2*z**Rational(3, 2)/3)/3) assert airyai(p).rewrite(besseli) == ( sqrt(p)*(besseli(-Rational(1, 3), 2*p**Rational(3, 2)/3) - besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3) assert expand_func(airyai(2*cbrt(3*z**5))) == ( -sqrt(3)*(-1 + cbrt(z**5)/z**Rational(5, 3))*airybi(2*cbrt(3)*z**Rational(5, 3))/6 + (1 + cbrt(z**5)/z**Rational(5, 3))*airyai(2*cbrt(3)*z**Rational(5, 3))/2) assert expand_func(airyai(x*y)) == airyai(x*y) assert expand_func(airyai(log(x))) == airyai(log(x)) assert expand_func(airyai(2*root(3*z**5, 5))) == airyai(2*root(3*z**5, 5)) assert (airyai(r).as_real_imag() == airyai(r).as_real_imag(deep=False) == (airyai(r), 0)) assert airyai(x).as_real_imag() == airyai(x).as_real_imag(deep=False) assert (airyai(x).as_real_imag() == (airyai(re(x) - I*re(x)*abs(im(x))/abs(re(x)))/2 + airyai(re(x) + I*re(x)*abs(im(x))/abs(re(x)))/2, I*(airyai(re(x) - I*re(x)*abs(im(x))/abs(re(x))) - airyai(re(x) + I*re(x)*abs(im(x))/Abs(re(x)))) * re(x)*abs(im(x))/(2*im(x)*abs(re(x))))) assert airyai(x).taylor_term(-1, x) == 0
def test_airybiprime(): z = Symbol('z', extended_real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybiprime(z), airybiprime) assert airybiprime(0) == root(3, 6) / gamma(Rational(1, 3)) assert airybiprime(oo) == oo assert airybiprime(-oo) == 0 assert diff(airybiprime(z), z) == z * airybi(z) assert series(airybiprime(z), z, 0, 3) == (root(3, 6) / gamma(Rational(1, 3)) + 3**Rational(5, 6) * z**2 / (6 * gamma(Rational(2, 3))) + O(z**3)) assert airybiprime(z).rewrite(hyper) == ( 3**Rational(5, 6) * z**2 * hyper((), (Rational(5, 3), ), z**3 / 9) / (6 * gamma(Rational(2, 3))) + root(3, 6) * hyper( (), (Rational(1, 3), ), z**3 / 9) / gamma(Rational(1, 3))) assert isinstance(airybiprime(z).rewrite(besselj), airybiprime) assert (airybiprime(t).rewrite(besselj) == -sqrt(3) * t * (besselj(-Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3) + besselj(Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3) assert airybiprime(z).rewrite(besseli) == ( sqrt(3) * (z**2 * besseli(Rational(2, 3), 2 * z**Rational(3, 2) / 3) / (z**Rational(3, 2))**Rational(2, 3) + (z**Rational(3, 2))**Rational(2, 3) * besseli(-Rational(2, 3), 2 * z**Rational(3, 2) / 3)) / 3) assert airybiprime(p).rewrite(besseli) == ( sqrt(3) * p * (besseli(-Rational(2, 3), 2 * p**Rational(3, 2) / 3) + besseli(Rational(2, 3), 2 * p**Rational(3, 2) / 3)) / 3) assert airybiprime(p).rewrite(besselj) == airybiprime(p) assert expand_func(airybiprime( 2 * cbrt(3 * z**5))) == (sqrt(3) * (z**Rational(5, 3) / cbrt(z**5) - 1) * airyaiprime(2 * cbrt(3) * z**Rational(5, 3)) / 2 + (z**Rational(5, 3) / cbrt(z**5) + 1) * airybiprime(2 * cbrt(3) * z**Rational(5, 3)) / 2) assert expand_func(airybiprime(x * y)) == airybiprime(x * y) assert expand_func(airybiprime(log(x))) == airybiprime(log(x)) assert expand_func(airybiprime(2 * root(3 * z**5, 5))) == airybiprime( 2 * root(3 * z**5, 5)) assert airybiprime(-2).evalf(50) == Float( '0.27879516692116952268509756941098324140300059345163131', dps=50)
def test_issue_5223(): assert series(1, x) == 1 assert next(Integer(0).lseries(x)) == 0 assert cos(x).series() == cos(x).series(x) pytest.raises(ValueError, lambda: cos(x + y).series()) pytest.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().removeO() == 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 assert ((sin(x))**y).nseries(x, n=1) == x**y + O(x**(y + 2), 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 + x**3*log(x)**3/6 + O(x**3) p = Symbol('p', positive=True) assert exp(sqrt(p)**3*log(p)).series(n=3) == \ 1 + p**3*log(p)**2/2 + p**Rational(3, 2)*log(p) + O(p**3) assert exp(sin(x)*log(x)).series(n=2) == \ 1 + x*log(x) + x**2*log(x)**2/2 + O(x**2)
def test_airyaiprime(): z = Symbol('z', extended_real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airyaiprime(z), airyaiprime) assert airyaiprime(0) == -3**Rational(2, 3) / (3 * gamma(Rational(1, 3))) assert airyaiprime(oo) == 0 assert diff(airyaiprime(z), z) == z * airyai(z) assert series(airyaiprime(z), z, 0, 3) == (-3**Rational(2, 3) / (3 * gamma(Rational(1, 3))) + cbrt(3) * z**2 / (6 * gamma(Rational(2, 3))) + O(z**3)) assert airyaiprime(z).rewrite(hyper) == ( cbrt(3) * z**2 * hyper((), (Rational(5, 3), ), z**3 / 9) / (6 * gamma(Rational(2, 3))) - 3**Rational(2, 3) * hyper( (), (Rational(1, 3), ), z**3 / 9) / (3 * gamma(Rational(1, 3)))) assert isinstance(airyaiprime(z).rewrite(besselj), airyaiprime) assert (airyaiprime(t).rewrite(besselj) == t * (besselj(-Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3) - besselj(Rational(2, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3) assert airyaiprime(z).rewrite(besseli) == ( z**2 * besseli(Rational(2, 3), 2 * z**Rational(3, 2) / 3) / (3 * (z**Rational(3, 2))**Rational(2, 3)) - (z**Rational(3, 2))**Rational(2, 3) * besseli(-Rational(1, 3), 2 * z**Rational(3, 2) / 3) / 3) assert airyaiprime(p).rewrite(besseli) == ( p * (-besseli(-Rational(2, 3), 2 * p**Rational(3, 2) / 3) + besseli(Rational(2, 3), 2 * p**Rational(3, 2) / 3)) / 3) assert airyaiprime(p).rewrite(besselj) == airyaiprime(p) assert expand_func(airyaiprime( 2 * cbrt(3 * z**5))) == (sqrt(3) * (z**Rational(5, 3) / cbrt(z**5) - 1) * airybiprime(2 * cbrt(3) * z**Rational(5, 3)) / 6 + (z**Rational(5, 3) / cbrt(z**5) + 1) * airyaiprime(2 * cbrt(3) * z**Rational(5, 3)) / 2) assert expand_func(airyaiprime(x * y)) == airyaiprime(x * y) assert expand_func(airyaiprime(log(x))) == airyaiprime(log(x)) assert expand_func(airyaiprime(2 * root(3 * z**5, 5))) == airyaiprime( 2 * root(3 * z**5, 5)) assert airyaiprime(-2).evalf(50) == Float( '0.61825902074169104140626429133247528291577794512414753', dps=50)
def test_sympyissue_5223(): assert series(1, x) == 1 assert next(Integer(0).lseries(x)) == 0 assert cos(x).series() == cos(x).series(x) pytest.raises(ValueError, lambda: cos(x + y).series()) pytest.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().removeO() == 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 assert ((sin(x))**y).nseries(x, n=1) == x**y + O(x**(y + 2), 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 + x**3*log(x)**3/6 + O(x**3) p = Symbol('p', positive=True) assert exp(sqrt(p)**3*log(p)).series(n=3) == \ 1 + p**3*log(p)**2/2 + p**Rational(3, 2)*log(p) + O(p**3) assert exp(sin(x)*log(x)).series(n=2) == \ 1 + x*log(x) + x**2*log(x)**2/2 + O(x**2)
def test_airyai(): z = Symbol('z', extended_real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airyai(z), airyai) assert airyai(0) == 3**Rational(1, 3) / (3 * gamma(Rational(2, 3))) assert airyai(oo) == 0 assert airyai(-oo) == 0 assert diff(airyai(z), z) == airyaiprime(z) assert series(airyai(z), z, 0, 3) == (3**Rational(5, 6) * gamma(Rational(1, 3)) / (6 * pi) - 3**Rational(1, 6) * z * gamma(Rational(2, 3)) / (2 * pi) + O(z**3)) assert airyai(z).rewrite(hyper) == (-3**Rational(2, 3) * z * hyper( (), (Rational(4, 3), ), z**Integer(3) / 9) / (3 * gamma(Rational(1, 3))) + 3**Rational(1, 3) * hyper( (), (Rational(2, 3), ), z**Integer(3) / 9) / (3 * gamma(Rational(2, 3)))) assert isinstance(airyai(z).rewrite(besselj), airyai) assert airyai(t).rewrite(besselj) == ( sqrt(-t) * (besselj(-Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3) + besselj(Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3) assert airyai(z).rewrite(besseli) == ( -z * besseli(Rational(1, 3), 2 * z**Rational(3, 2) / 3) / (3 * (z**Rational(3, 2))**Rational(1, 3)) + (z**Rational(3, 2))**Rational(1, 3) * besseli(-Rational(1, 3), 2 * z**Rational(3, 2) / 3) / 3) assert airyai(p).rewrite(besseli) == ( sqrt(p) * (besseli(-Rational(1, 3), 2 * p**Rational(3, 2) / 3) - besseli(Rational(1, 3), 2 * p**Rational(3, 2) / 3)) / 3) assert expand_func(airyai(2 * (3 * z**5)**Rational(1, 3))) == ( -sqrt(3) * (-1 + (z**5)**Rational(1, 3) / z**Rational(5, 3)) * airybi(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 6 + (1 + (z**5)**Rational(1, 3) / z**Rational(5, 3)) * airyai(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 2)
def test_airybi(): z = Symbol('z', extended_real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybi(z), airybi) assert airybi(0) == 3**Rational(5, 6)/(3*gamma(Rational(2, 3))) assert airybi(oo) == oo assert airybi(-oo) == 0 assert diff(airybi(z), z) == airybiprime(z) assert series(airybi(z), z, 0, 3) == ( cbrt(3)*gamma(Rational(1, 3))/(2*pi) + 3**Rational(2, 3)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3)) l = Limit(airybi(I/x)/(exp(Rational(2, 3)*(I/x)**Rational(3, 2))*sqrt(pi*sqrt(I/x))), x, 0) assert l.doit() == l assert airybi(z).rewrite(hyper) == ( root(3, 6)*z*hyper((), (Rational(4, 3),), z**3/9)/gamma(Rational(1, 3)) + 3**Rational(5, 6)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3)))) assert isinstance(airybi(z).rewrite(besselj), airybi) assert (airybi(t).rewrite(besselj) == sqrt(3)*sqrt(-t)*(besselj(-1/3, 2*(-t)**Rational(3, 2)/3) - besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3) assert airybi(z).rewrite(besseli) == ( sqrt(3)*(z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/cbrt(z**Rational(3, 2)) + cbrt(z**Rational(3, 2))*besseli(-Rational(1, 3), 2*z**Rational(3, 2)/3))/3) assert airybi(p).rewrite(besseli) == ( sqrt(3)*sqrt(p)*(besseli(-Rational(1, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3) assert airybi(p).rewrite(besselj) == airybi(p) assert expand_func(airybi(2*cbrt(3*z**5))) == ( sqrt(3)*(1 - cbrt(z**5)/z**Rational(5, 3))*airyai(2*cbrt(3)*z**Rational(5, 3))/2 + (1 + cbrt(z**5)/z**Rational(5, 3))*airybi(2*cbrt(3)*z**Rational(5, 3))/2) assert expand_func(airybi(x*y)) == airybi(x*y) assert expand_func(airybi(log(x))) == airybi(log(x)) assert expand_func(airybi(2*root(3*z**5, 5))) == airybi(2*root(3*z**5, 5)) assert airybi(x).taylor_term(-1, x) == 0
def test_airybiprime(): z = Symbol('z', extended_real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybiprime(z), airybiprime) assert airybiprime(0) == root(3, 6)/gamma(Rational(1, 3)) assert airybiprime(oo) == oo assert airybiprime(-oo) == 0 assert diff(airybiprime(z), z) == z*airybi(z) assert series(airybiprime(z), z, 0, 3) == ( root(3, 6)/gamma(Rational(1, 3)) + 3**Rational(5, 6)*z**2/(6*gamma(Rational(2, 3))) + O(z**3)) assert airybiprime(z).rewrite(hyper) == ( 3**Rational(5, 6)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) + root(3, 6)*hyper((), (Rational(1, 3),), z**3/9)/gamma(Rational(1, 3))) assert isinstance(airybiprime(z).rewrite(besselj), airybiprime) assert (airybiprime(t).rewrite(besselj) == -sqrt(3)*t*(besselj(-Rational(2, 3), 2*(-t)**Rational(3, 2)/3) + besselj(Rational(2, 3), 2*(-t)**Rational(3, 2)/3))/3) assert airybiprime(z).rewrite(besseli) == ( sqrt(3)*(z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(z**Rational(3, 2))**Rational(2, 3) + (z**Rational(3, 2))**Rational(2, 3)*besseli(-Rational(2, 3), 2*z**Rational(3, 2)/3))/3) assert airybiprime(p).rewrite(besseli) == ( sqrt(3)*p*(besseli(-Rational(2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3) assert airybiprime(p).rewrite(besselj) == airybiprime(p) assert expand_func(airybiprime(2*cbrt(3*z**5))) == ( sqrt(3)*(z**Rational(5, 3)/cbrt(z**5) - 1)*airyaiprime(2*cbrt(3)*z**Rational(5, 3))/2 + (z**Rational(5, 3)/cbrt(z**5) + 1)*airybiprime(2*cbrt(3)*z**Rational(5, 3))/2) assert expand_func(airybiprime(x*y)) == airybiprime(x*y) assert expand_func(airybiprime(log(x))) == airybiprime(log(x)) assert expand_func(airybiprime(2*root(3*z**5, 5))) == airybiprime(2*root(3*z**5, 5)) assert airybiprime(-2).evalf(50) == Float('0.27879516692116952268509756941098324140300059345163131', dps=50)
def test_airyaiprime(): z = Symbol('z', extended_real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airyaiprime(z), airyaiprime) assert airyaiprime(0) == -3**Rational(2, 3)/(3*gamma(Rational(1, 3))) assert airyaiprime(oo) == 0 assert diff(airyaiprime(z), z) == z*airyai(z) assert series(airyaiprime(z), z, 0, 3) == ( -3**Rational(2, 3)/(3*gamma(Rational(1, 3))) + cbrt(3)*z**2/(6*gamma(Rational(2, 3))) + O(z**3)) assert airyaiprime(z).rewrite(hyper) == ( cbrt(3)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) - 3**Rational(2, 3)*hyper((), (Rational(1, 3),), z**3/9)/(3*gamma(Rational(1, 3)))) assert isinstance(airyaiprime(z).rewrite(besselj), airyaiprime) assert (airyaiprime(t).rewrite(besselj) == t*(besselj(-Rational(2, 3), 2*(-t)**Rational(3, 2)/3) - besselj(Rational(2, 3), 2*(-t)**Rational(3, 2)/3))/3) assert airyaiprime(z).rewrite(besseli) == ( z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(3*(z**Rational(3, 2))**Rational(2, 3)) - (z**Rational(3, 2))**Rational(2, 3)*besseli(-Rational(1, 3), 2*z**Rational(3, 2)/3)/3) assert airyaiprime(p).rewrite(besseli) == ( p*(-besseli(-Rational(2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3) assert airyaiprime(p).rewrite(besselj) == airyaiprime(p) assert expand_func(airyaiprime(2*cbrt(3*z**5))) == ( sqrt(3)*(z**Rational(5, 3)/cbrt(z**5) - 1)*airybiprime(2*cbrt(3)*z**Rational(5, 3))/6 + (z**Rational(5, 3)/cbrt(z**5) + 1)*airyaiprime(2*cbrt(3)*z**Rational(5, 3))/2) assert expand_func(airyaiprime(x*y)) == airyaiprime(x*y) assert expand_func(airyaiprime(log(x))) == airyaiprime(log(x)) assert expand_func(airyaiprime(2*root(3*z**5, 5))) == airyaiprime(2*root(3*z**5, 5)) assert airyaiprime(-2).evalf(50) == Float('0.61825902074169104140626429133247528291577794512414753', dps=50)
def test_issue_7203(): assert series(cos(x), x, pi, 3) == \ -1 + (x - pi)**2/2 + O((x - pi)**3, (x, pi))
def test_airyai(): z = Symbol('z', extended_real=False) r = Symbol('r', extended_real=True) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airyai(z), airyai) assert airyai(0) == cbrt(3) / (3 * gamma(Rational(2, 3))) assert airyai(oo) == 0 assert airyai(-oo) == 0 assert diff(airyai(z), z) == airyaiprime(z) assert series(airyai(z), z, 0, 3) == (3**Rational(5, 6) * gamma(Rational(1, 3)) / (6 * pi) - root(3, 6) * z * gamma(Rational(2, 3)) / (2 * pi) + O(z**3)) l = Limit( airyai(I / x) / (exp(-Rational(2, 3) * (I / x)**Rational(3, 2)) * sqrt(pi * sqrt(I / x)) / 2), x, 0) assert l.doit() == l # cover _airyais._eval_aseries assert airyai(z).rewrite(hyper) == (-3**Rational(2, 3) * z * hyper( (), (Rational(4, 3), ), z**3 / 9) / (3 * gamma(Rational(1, 3))) + cbrt(3) * hyper( (), (Rational(2, 3), ), z**3 / 9) / (3 * gamma(Rational(2, 3)))) assert isinstance(airyai(z).rewrite(besselj), airyai) assert airyai(t).rewrite(besselj) == ( sqrt(-t) * (besselj(-Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3) + besselj(Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3) assert airyai(z).rewrite(besseli) == ( -z * besseli(Rational(1, 3), 2 * z**Rational(3, 2) / 3) / (3 * cbrt(z**Rational(3, 2))) + cbrt(z**Rational(3, 2)) * besseli(-Rational(1, 3), 2 * z**Rational(3, 2) / 3) / 3) assert airyai(p).rewrite(besseli) == ( sqrt(p) * (besseli(-Rational(1, 3), 2 * p**Rational(3, 2) / 3) - besseli(Rational(1, 3), 2 * p**Rational(3, 2) / 3)) / 3) assert expand_func(airyai( 2 * cbrt(3 * z**5))) == (-sqrt(3) * (-1 + cbrt(z**5) / z**Rational(5, 3)) * airybi(2 * cbrt(3) * z**Rational(5, 3)) / 6 + (1 + cbrt(z**5) / z**Rational(5, 3)) * airyai(2 * cbrt(3) * z**Rational(5, 3)) / 2) assert expand_func(airyai(x * y)) == airyai(x * y) assert expand_func(airyai(log(x))) == airyai(log(x)) assert expand_func(airyai(2 * root(3 * z**5, 5))) == airyai( 2 * root(3 * z**5, 5)) assert (airyai(r).as_real_imag() == airyai(r).as_real_imag(deep=False) == (airyai(r), 0)) assert airyai(x).as_real_imag() == airyai(x).as_real_imag(deep=False) assert (airyai(x).as_real_imag() == ( airyai(re(x) - I * re(x) * abs(im(x)) / abs(re(x))) / 2 + airyai(re(x) + I * re(x) * abs(im(x)) / abs(re(x))) / 2, I * (airyai(re(x) - I * re(x) * abs(im(x)) / abs(re(x))) - airyai(re(x) + I * re(x) * abs(im(x)) / Abs(re(x)))) * re(x) * abs(im(x)) / (2 * im(x) * abs(re(x))))) assert airyai(x).taylor_term(-1, x) == 0
def test_sin(): e1 = sin(x).series(x, 0) e2 = series(sin(x), x, 0) assert e1 == e2
def test_csc(): x = symbols('x', extended_real=True) z = symbols('z') # https://github.com/sympy/sympy/issues/6707 cosecant = csc('x') alternate = 1/sin('x') assert cosecant.equals(alternate) assert alternate.equals(cosecant) assert csc.nargs == FiniteSet(1) assert csc(0) == zoo assert csc(pi) == zoo assert csc(pi/2) == 1 assert csc(-pi/2) == -1 assert csc(pi/6) == 2 assert csc(pi/3) == 2*sqrt(3)/3 assert csc(5*pi/2) == 1 assert csc(9*pi/7) == -csc(2*pi/7) assert csc(3*pi/4) == sqrt(2) # issue sympy/sympy#8421 assert csc(I) == -I*csch(1) assert csc(x*I) == -I*csch(x) assert csc(-x) == -csc(x) assert csc(acsc(x)) == x assert csc(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x)) assert csc(x).rewrite(sin) == 1/sin(x) assert csc(x).rewrite(cos) == csc(x) assert csc(x).rewrite(tan) == (tan(x/2)**2 + 1)/(2*tan(x/2)) assert csc(x).rewrite(cot) == (cot(x/2)**2 + 1)/(2*cot(x/2)) assert csc(z).conjugate() == csc(conjugate(z)) assert (csc(z).as_real_imag() == (sin(re(z))*cosh(im(z))/(sin(re(z))**2*cosh(im(z))**2 + cos(re(z))**2*sinh(im(z))**2), -cos(re(z))*sinh(im(z))/(sin(re(z))**2*cosh(im(z))**2 + cos(re(z))**2*sinh(im(z))**2))) assert csc(x).expand(trig=True) == 1/sin(x) assert csc(2*x).expand(trig=True) == 1/(2*sin(x)*cos(x)) assert csc(a).is_algebraic is None assert csc(na).is_algebraic is False assert csc(x).as_leading_term() == csc(x) assert csc(0).is_finite is False assert csc(x).is_finite is None assert csc(pi/2).is_finite assert series(csc(x), x, x0=pi/2, n=6) == \ 1 + (x - pi/2)**2/2 + 5*(x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2)) assert series(csc(x), x, x0=0, n=6) == \ 1/x + x/6 + 7*x**3/360 + 31*x**5/15120 + O(x**6) assert csc(x).diff(x) == -cot(x)*csc(x) assert csc(x).taylor_term(0, x) == 1/x assert csc(x).taylor_term(2, x) == 0 assert csc(x).taylor_term(3, x) == 7*x**3/360 assert csc(x).taylor_term(5, x) == 31*x**5/15120 pytest.raises(ArgumentIndexError, lambda: csc(x).fdiff(2))
def test_exp2(): e1 = exp(cos(x)).series(x, 0) e2 = series(exp(cos(x)), x, 0) assert e1 == e2
def test_sympyissue_15539(): assert series(exp(x), x, x0=-oo) == exp(-x)
def test_sympyissue_8805(): assert series(1, n=8) == 1
def test_sympyissue_7203(): assert series(cos(x), x, pi, 3) == \ -1 + (x - pi)**2/2 + O((x - pi)**3, (x, pi))
def test_sympyissue_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_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_sympyissue_15539(): assert series(exp(x), x, x0=-oo) == exp(-x)
def test_exp_product_positive_factors(): a, b = symbols('a, b', positive=True) x = a * b series(exp(x), x, n=8)
def test_sin(): e1 = sin(x).series(x, 0) e2 = series(sin(x), x, 0) assert e1 == e2
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)
def test_exp_product_positive_factors(): a, b = symbols('a, b', positive=True) x = a * b series(exp(x), x, n=8)
def test_issue_8805(): assert series(1, n=8) == 1
def test_cos(): e1 = cos(x).series(x, 0) e2 = series(cos(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_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_sec(): x = symbols('x', extended_real=True) z = symbols('z') assert sec.nargs == FiniteSet(1) assert sec(0) == 1 assert sec(pi) == -1 assert sec(pi/2) == zoo assert sec(-pi/2) == zoo assert sec(pi/6) == 2*sqrt(3)/3 assert sec(pi/3) == 2 assert sec(5*pi/2) == zoo assert sec(9*pi/7) == -sec(2*pi/7) assert sec(3*pi/4) == -sqrt(2) # issue sympy/sympy#8421 assert sec(I) == sech(1) assert sec(x*I) == sech(x) assert sec(-x) == sec(x) assert sec(asec(x)) == x assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2) assert sec(x).rewrite(sin) == sec(x) assert sec(x).rewrite(cos) == 1/cos(x) assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1) assert sec(x).rewrite(sqrt) == sec(x) assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1) assert sec(z).conjugate() == sec(conjugate(z)) assert (sec(z).as_real_imag() == (cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 + cos(re(z))**2*cosh(im(z))**2), sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 + cos(re(z))**2*cosh(im(z))**2))) assert sec(x).expand(trig=True) == 1/cos(x) assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1) assert sec(a).is_algebraic is None assert sec(na).is_algebraic is False assert sec(x).as_leading_term() == sec(x) assert sec(0).is_finite assert sec(x).is_finite is None assert sec(pi/2).is_finite is False assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6) # https://github.com/sympy/sympy/issues/7166 assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6) # https://github.com/sympy/sympy/issues/7167 assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) == 1/sqrt(x - 3*pi/2) + (x - 3*pi/2)**Rational(3, 2)/12 + (x - 3*pi/2)**Rational(7, 2)/160 + O((x - 3*pi/2)**4, (x, 3*pi/2))) assert sec(x).diff(x) == tan(x)*sec(x) # Taylor Term checks assert sec(z).taylor_term(4, z) == 5*z**4/24 assert sec(z).taylor_term(6, z) == 61*z**6/720 assert sec(z).taylor_term(5, z) == 0 pytest.raises(ArgumentIndexError, lambda: sec(x).fdiff(2))