def test_Limit():
assert str(Limit(sin(x) / x, x, y)) == 'Limit(sin(x)/x, x, y)'
assert str(Limit(1 / x, x, 0)) == 'Limit(1/x, x, 0)'
assert str(Limit(sin(x) / x, x, y,
dir=1)) == 'Limit(sin(x)/x, x, y, dir=1)'
assert str(Limit(sin(x) / x, x, y,
dir=Reals)) == 'Limit(sin(x)/x, x, y, dir=Reals)'
def test__eis():
assert _eis(z).diff(z) == -_eis(z) + 1 / z
pytest.raises(ArgumentIndexError, lambda: _eis(x).fdiff(2))
assert _eis(1/z).series(z) == \
z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
assert Ei(z).rewrite('tractable') == exp(z) * _eis(z)
assert li(z).rewrite('tractable') == z * _eis(log(z))
assert _eis(z).rewrite('intractable') == exp(-z) * Ei(z)
assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== li(z).diff(z)
assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== Ei(z).diff(z)
assert _eis(z).series(
z,
n=2) == EulerGamma + log(z) + z * (-log(z) - EulerGamma + 1) + z**2 * (
log(z) / 2 - Rational(3, 4) + EulerGamma / 2) + O(z**2)
l = Limit(Ei(y / x) / exp(y / x), x, 0)
assert l.doit() == l # cover _eis._eval_aseries
def test_Limit():
e = Limit(sin(x)/x, x, 0)
assert mathematica_code(e) == 'Hold[Limit[Sin[x]/x, x -> 0, Direction -> -1]]'
e = Limit(sin(x)/x, x, 0, '-')
assert mathematica_code(e) == 'Hold[Limit[Sin[x]/x, x -> 0, Direction -> 1]]'
e = Limit(sin(x)/x, x, 0, 'real')
assert mathematica_code(e) == 'Hold[Limit[Sin[x]/x, x -> 0, Direction -> Reals]]'
def test_basic4():
assert limit(2*x + y*x, x, 0) == 0
assert limit(2*x + y*x, x, 1) == 2 + y
assert limit(2*x**8 + y*x**(-3), x, -2) == 512 - y/8
assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0
assert integrate(1/(x**3 + 1), (x, 0, oo)) == 2*pi*sqrt(3)/9
# coverage test
l = Limit(Piecewise((x, x > 1), (0, True)), x, -1)
assert l.doit() == l
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 airyai(z).series(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_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_diofant_parser():
x = Symbol('x')
inputs = {
'2*x':
2 * x,
'3.00':
Float(3),
'22/7':
Rational(22, 7),
'2+3j':
2 + 3 * I,
'exp(x)':
exp(x),
'-(2)':
-Integer(2),
'[-1, -2, 3]': [Integer(-1), Integer(-2),
Integer(3)],
'Symbol("x").free_symbols':
x.free_symbols,
'Float(Integer(3).evalf(3))':
3.00,
'factorint(12, visual=True)':
Mul(Pow(2, 2, evaluate=False),
Pow(3, 1, evaluate=False),
evaluate=False),
'Limit(sin(x), x, 0, dir=1)':
Limit(sin(x), x, 0, dir=1),
}
for text, result in inputs.items():
assert parse_expr(text) == result
def test_mathml_limits():
# XXX No unevaluated limits
lim_fun = sin(x) / x
mml_1 = mp._print(Limit(lim_fun, x, 0))
assert mml_1.childNodes[0].nodeName == 'limit'
assert mml_1.childNodes[1].nodeName == 'bvar'
assert mml_1.childNodes[2].nodeName == 'lowlimit'
assert mml_1.childNodes[3].toxml() == mp._print(lim_fun).toxml()
def test_diofantissue_558():
n = Symbol('n')
r = Symbol('r', positive=True)
c = Symbol('c')
expr = ((2*n*(n - r + 1)/(n + r*(n - r + 1)))**c +
(r - 1)*(n*(n - r + 2)/(n + r*(n - r + 1)))**c - n)/(n**c - n)
expr = expr.subs({c: c + 1})
assert limit(expr, n, oo) == Limit(expr, n, oo)
def test_basic5():
class my(Function):
@classmethod
def eval(cls, arg):
if arg is S.Infinity:
return S.NaN
assert limit(my(x), x, oo) == Limit(my(x), x, oo)
def test_basic5():
class my(Function):
@classmethod
def eval(cls, arg):
if arg is S.Infinity:
return S.NaN
assert limit(my(x), x, oo) == Limit(my(x), x, oo)
assert limit(4/x > 8, x, 0) # relational test
assert limit(my(x) > 0, x, oo) == Limit(my(x) > 0, x, oo)
# from https://groups.google.com/forum/#!topic/sympy/LkTMQKC_BOw
# fix bisected to ade6d20 and c459d18
a = Symbol('a', positive=True)
f = exp(x*(-a - 1)) * sinh(x)
assert limit(f, x, oo) == 0
assert limit(O(x), x, x**2) == Limit(O(x), x, x**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_basic4():
assert limit(2*x + y*x, x, 0) == 0
assert limit(2*x + y*x, x, 1) == 2 + y
assert limit(2*x**8 + y*x**(-3), x, -2) == 512 - y/8
assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0
assert integrate(1/(x**3 + 1), (x, 0, oo)) == 2*pi*sqrt(3)/9
# coverage test
assert limit(Piecewise((x, x > 1), (0, True)), x, -1) == 0
# issue sympy/sympy#16714
e = ((n**(n + 1) + (n + 1)**n)/n**(n + 1))**n
assert limit(e, n, oo) == E**E
# issue sympy/sympy#18492
e1 = 2*sqrt(x)*Piecewise(((4*x - 2)/abs(sqrt(4 - 4*(2*x - 1)**2)),
4*x - 2 >= 0),
((2 - 4*x)/abs(sqrt(4 - 4*(2*x - 1)**2)), True))
e2 = Piecewise((x**2/2, x <= Rational(1, 2)), (x/2 - Rational(1, 8), True))
e3 = Piecewise(((x - 9)/5, x < -1), ((x - 9)/5, x > 4),
(sqrt(abs(x - 3)), True))
assert limit(e1, x, 0) == 1
assert limit(e2, x, 0) == 0
assert limit(e2, x, oo) == oo
assert limit(e3, x, -1) == 2
assert limit(e3, x, oo) == oo
e4 = Piecewise((1, 0 < x), (0, True))
assert limit(e4, x, 0, 1) == 0
assert limit(e4, x, 0) == 1
pytest.raises(PoleError, lambda: limit(e4, x, 0, Reals))
e5 = Piecewise((1, 0 < x), (2, 1 < x), (0, True))
assert limit(e5, x, oo) == 1
assert limit(e5, x, 1, 1) == 1
assert limit(e5, x, 1) == 1
e6 = Limit(Piecewise((1, x > a), (0, True)), x, 0)
assert e6.doit() == e6
def test_basic5():
class MyFunction(Function):
@classmethod
def eval(cls, arg):
if arg is oo:
return nan
assert limit(MyFunction(x), x, oo) == Limit(MyFunction(x), x, oo)
assert limit(4/x > 8, x, 0) is true # relational test
assert limit(MyFunction(x) > 0, x, oo) == Limit(MyFunction(x) > 0, x, oo)
# issue diofant/diofant#1217
assert limit(x > 0, x, 0) is true
assert limit(x > 0, x, 0, 1) is false
# issue sympy/sympy#11833
a = Symbol('a', positive=True)
f = exp(x*(-a - 1)) * sinh(x)
assert limit(f, x, oo) == 0
assert limit(O(x), x, x**2) == Limit(O(x), x, x**2)
def test__eis():
assert _eis(z).diff(z) == -_eis(z) + 1/z
pytest.raises(ArgumentIndexError, lambda: _eis(x).fdiff(2))
assert _eis(1/z).series(z) == \
z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
assert Ei(z).rewrite('tractable') == exp(z)*_eis(z)
assert li(z).rewrite('tractable') == z*_eis(log(z))
assert _eis(z).rewrite('intractable') == exp(-z)*Ei(z)
assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== li(z).diff(z)
assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== Ei(z).diff(z)
assert _eis(z).series(z, n=2) == EulerGamma + log(z) + z*(-log(z) -
EulerGamma + 1) + z**2*(log(z)/2 - Rational(3, 4) + EulerGamma/2) + O(z**2)
l = Limit(Ei(y/x)/exp(y/x), x, 0)
assert l.doit() == l # cover _eis._eval_aseries
def test_sympyissue_15673():
p = symbols('p')
alpha = symbols('α', positive=True)
e = Limit(4*pi*p**(-alpha)*(p**3 - p**alpha)/(alpha - 3), p, 0)
assert isinstance(e.doit(), Limit) # but see diofant/diofant#425
def test_series():
for c in (Limit, Limit(y, x, 1), Order, Order(y)):
check(c)
def test_sympyissue_7391():
f = Function('f')
func = x*y*y/(x*x + y**4)
l = Limit(func.subs({y: f(x)}), x, 0)
assert l.doit() == l
assert l.subs({f: Lambda(x, sqrt(x))}).doit() == Rational(1, 2)
def test_doit2():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
# limit() breaks on the contained Integral.
assert l.doit(deep=False) == l
def test_basic1():
assert limit(x, x, -oo) == -oo
assert limit(x**2, x, -oo) == oo
assert limit(x*log(x), x, 0) == 0
assert limit(x - x**2, x, oo) == -oo
assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
assert limit((1 + x)**oo, x, 0) == oo
assert limit((1 + x)**oo, x, 0, dir=1) == 0
assert limit((1 + x + y)**oo, x, 0, dir=1) == (1 + y)**oo
assert limit(y/x/log(x), x, 0) == -oo*sign(y)
assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo
limit(Sum(1/x, (x, 1, y)) - log(y), y, oo)
limit(Sum(1/x, (x, 1, y)) - 1/y, y, oo)
assert limit(nan, x, -oo) == nan
assert limit(O(2)*x, x, nan) == nan
assert limit(sin(O(x)), x, 0) == 0
assert limit(1/(x - 1), x, 1) == oo
assert limit(1/(x - 1), x, 1, dir=1) == -oo
assert limit(1/(5 - x)**3, x, 5) == -oo
assert limit(1/(5 - x)**3, x, 5, dir=1) == oo
assert limit(1/sin(x), x, pi) == -oo
assert limit(1/sin(x), x, pi, dir=1) == oo
assert limit(1/cos(x), x, pi/2) == -oo
assert limit(1/cos(x), x, pi/2, dir=1) == oo
assert limit(1/tan(x**3), x, cbrt(2*pi)) == oo
assert limit(1/tan(x**3), x, cbrt(2*pi), dir=1) == -oo
assert limit(1/cot(x)**3, x, 3*pi/2) == -oo
assert limit(1/cot(x)**3, x, 3*pi/2, dir=1) == oo
# approaching 0
# from dir=-1
assert limit(1 + 1/x, x, 0) == oo
# from dir=1
# Add
assert limit(1 + 1/x, x, 0, dir=1) == -oo
# Pow
assert limit(x**(-2), x, 0, dir=1) == oo
assert limit(x**(-3), x, 0, dir=1) == -oo
assert limit(1/sqrt(x), x, 0, dir=1) == (-oo)*I
assert limit(x**2, x, 0, dir=1) == 0
assert limit(sqrt(x), x, 0, dir=1) == 0
assert limit(x**-pi, x, 0, dir=1) == oo*sign((-1)**(-pi))
assert limit((1 + cos(x))**oo, x, 0) == oo
assert limit(x**2, x, 0, dir=Reals) == 0
assert limit(exp(x), x, 0, dir=Reals) == 1
pytest.raises(PoleError, lambda: limit(1/x, x, 0, dir=Reals))
# issue diofant/diofant#74
assert limit(sign(log(1 - 1/x)), x, oo) == -1
# issue sympy/sympy#8166
f = Function('f')
assert limit(f(x), x, 4) == Limit(f(x), x, 4)
assert limit(exp(x), x, 0, dir=exp(I*pi/3)) == 1
assert limit(sqrt(-1 + I*x), x, 0) == +I
assert limit(sqrt(-1 + I*x), x, 0, dir=1) == -I
assert limit(sqrt(-1 + I*x), x, 0, dir=exp(I*pi/3)) == -I
assert limit(log(x + sqrt(x**2 + 1)), x, I) == I*pi/2
assert limit(log(x + sqrt(x**2 + 1)), x, I, dir=1) == I*pi/2
assert limit(log(x + sqrt(x**2 + 1)), x, I, dir=exp(I*pi/3)) == I*pi/2
def test_erf():
assert erf(nan) == nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) == 0
assert erf(I*oo) == oo*I
assert erf(-I*oo) == -oo*I
assert erf(-2) == -erf(2)
assert erf(-x*y) == -erf(x*y)
assert erf(-x - y) == -erf(x + y)
assert erf(erfinv(x)) == x
assert erf(erfcinv(x)) == 1 - x
assert erf(erf2inv(0, x)) == x
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_extended_real is False
assert erf(w).is_extended_real is True
assert erf(z).is_extended_real is None
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2*x/sqrt(pi)
assert erf(1/x).as_leading_term(x) == erf(1/x)
assert erf(z).rewrite('uppergamma') == sqrt(z**2)*erf(sqrt(z**2))/z
assert erf(z).rewrite('erfc') == 1 - erfc(z)
assert erf(z).rewrite('erfi') == -I*erfi(I*z)
assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erf(z).rewrite('hyper') == 2*z*hyper([Rational(1, 2)], [Rational(3, 2)], -z**2)/sqrt(pi)
assert erf(z).rewrite('meijerg') == z*meijerg([Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi)
assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(Rational(1, 2), z**2)/sqrt(pi)
assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
2/sqrt(pi)
assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi)
assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1
assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1
l = Limit((1 - erf(y/x))*exp(y**2/x**2), x, 0)
assert l.doit() == l # cover _erfs._eval_aseries
assert erf(x).as_real_imag() == \
((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
assert erf(x).as_real_imag() == erf(x).as_real_imag(deep=False)
assert erf(w).as_real_imag() == (erf(w), 0)
assert erf(w).as_real_imag() == erf(w).as_real_imag(deep=False)
assert erf(I).as_real_imag() == (0, erfi(1))
pytest.raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
assert erf(x).taylor_term(3, x, *(2*x/sqrt(pi), 0)) == -2*x**3/3/sqrt(pi)
def test_doit2():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
# limit() breaks on the contained Integral.
assert l.doit(deep=False) == l
def test_doit():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
assert l.doit() == oo
def test_Limit():
assert Limit(sin(x)/x, x, 0) != 1
assert Limit(sin(x)/x, x, 0).doit() == 1
def test_doit():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
assert l.doit() == oo
def test_sympyissue_9205():
assert Limit(x, x, a).free_symbols == {a}
assert Limit(x, x, a, '-').free_symbols == {a}
assert Limit(x + y, x + y, a).free_symbols == {a}
assert Limit(-x**2 + y, x**2, a).free_symbols == {y, a}
def test_Limit_dir():
pytest.raises(TypeError, lambda: Limit(x, x, 0, dir=0))
pytest.raises(ValueError, lambda: Limit(x, x, 0, dir='0'))
def test_issue_1164():
# also sympy/sympy#14502
assert limit(factorial(x) - x**x, x, oo) == -oo
l = Limit(factorial(x) - x**oo, x, oo)
assert l.doit() == l
def test_erf():
assert erf(nan) == nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) == 0
assert erf(I * oo) == oo * I
assert erf(-I * oo) == -oo * I
assert erf(-2) == -erf(2)
assert erf(-x * y) == -erf(x * y)
assert erf(-x - y) == -erf(x + y)
assert erf(erfinv(x)) == x
assert erf(erfcinv(x)) == 1 - x
assert erf(erf2inv(0, x)) == x
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_extended_real is False
assert erf(w).is_extended_real is True
assert erf(z).is_extended_real is None
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi)
assert erf(1 / x).as_leading_term(x) == erf(1 / x)
assert erf(z).rewrite('uppergamma') == sqrt(z**2) * erf(sqrt(z**2)) / z
assert erf(z).rewrite('erfc') == 1 - erfc(z)
assert erf(z).rewrite('erfi') == -I * erfi(I * z)
assert erf(z).rewrite('fresnels') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('fresnelc') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('hyper') == 2 * z * hyper(
[Rational(1, 2)], [Rational(3, 2)], -z**2) / sqrt(pi)
assert erf(z).rewrite('meijerg') == z * meijerg(
[Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2) / sqrt(pi)
assert erf(z).rewrite('expint') == sqrt(z**2) / z - z * expint(
Rational(1, 2), z**2) / sqrt(pi)
assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
2/sqrt(pi)
assert limit((1 - erf(z)) * exp(z**2) * z, z, oo) == 1 / sqrt(pi)
assert limit((1 - erf(x)) * exp(x**2) * sqrt(pi) * x, x, oo) == 1
assert limit(((1 - erf(x)) * exp(x**2) * sqrt(pi) * x - 1) * 2 * x**2, x,
oo) == -1
l = Limit((1 - erf(y / x)) * exp(y**2 / x**2), x, 0)
assert l.doit() == l # cover _erfs._eval_aseries
assert erf(x).as_real_imag() == \
((erf(re(x) - I*re(x)*abs(im(x))/abs(re(x)))/2 +
erf(re(x) + I*re(x)*abs(im(x))/abs(re(x)))/2,
I*(erf(re(x) - I*re(x)*abs(im(x))/abs(re(x))) -
erf(re(x) + I*re(x)*abs(im(x))/abs(re(x)))) *
re(x)*abs(im(x))/(2*im(x)*abs(re(x)))))
assert erf(x).as_real_imag() == erf(x).as_real_imag(deep=False)
assert erf(w).as_real_imag() == (erf(w), 0)
assert erf(w).as_real_imag() == erf(w).as_real_imag(deep=False)
assert erf(I).as_real_imag() == (0, erfi(1))
pytest.raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
assert erf(x).taylor_term(3, x, *(2 * x / sqrt(pi),
0)) == -2 * x**3 / 3 / sqrt(pi)
def test_Limit_free_symbols():
# issue sympy/sympy#9205
assert Limit(x, x, a).free_symbols == {a}
assert Limit(x, x, a, 1).free_symbols == {a}
assert Limit(x + y, x + y, a).free_symbols == {a}
assert Limit(-x**2 + y, x**2, a).free_symbols == {y, a}
def test_Limit():
assert str(Limit(sin(x) / x, x, y)) == "Limit(sin(x)/x, x, y)"
assert str(Limit(1 / x, x, 0)) == "Limit(1/x, x, 0)"
assert str(Limit(sin(x) / x, x, y,
dir="-")) == "Limit(sin(x)/x, x, y, dir='-')"
def test_Limit():
assert str(Limit(sin(x) / x, x, y)) == 'Limit(sin(x)/x, x, y)'
assert str(Limit(1 / x, x, 0)) == 'Limit(1/x, x, 0)'
assert str(Limit(sin(x) / x, x, y,
dir='-')) == "Limit(sin(x)/x, x, y, dir='-')"