def test_Limits_simple_1():
assert limit((x + 1) * (x + 2) * (x + 3) / x**3, x, oo) == 1 # 172
assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0 # 179
assert limit((2 * x - 3) * (3 * x + 5) * (4 * x - 6) /
(3 * x**3 + x - 1), x, oo) == 8 # Primjer 1
assert limit(x / root3(x**3 + 10), x, oo) == 1 # Primjer 2
assert limit((x + 1)**2 / (x**2 + 1), x, oo) == 1 # 181
def test_Limits_simple_4a():
assert limit((sqrt(x) - sqrt(a)) / (x - a), x,
a) == 1 / (2 * sqrt(a)) # Primer 5
assert limit((sqrt(x) - 1) / (root3(x) - 1), x,
1) == Rational(3) / 2 # 205
assert limit((sqrt(1 + x) - sqrt(1 - x)) / x, x, 0) == 1 # 207
assert limit(sqrt(x**2 - 5 * x + 6) - x, x, oo) == -Rational(5) / 2 # 213
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') == S.One - 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([S.Half], [3 * S.Half],
-z**2) / sqrt(pi)
assert erf(z).rewrite('meijerg') == z * meijerg([S.Half], [], [0],
[-S.Half], z**2) / sqrt(pi)
assert erf(z).rewrite(
'expint') == sqrt(z**2) / z - z * expint(S.Half, z**2) / sqrt(S.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
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)))))
pytest.raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
def test_sympyissue_18176():
x = Symbol('x', real=True, positive=True)
n = Symbol('n', integer=True, positive=True)
k = Symbol('k')
e = x**n - x**(n - k)
assert limit(e.subs({k: 0}), x, oo) == 0
assert limit(e.subs({k: 1}), x, oo) == oo
def test_harmonic_limit():
n = Symbol("n")
m = Symbol("m", positive=True)
assert limit(harmonic(n, m + 1), n, oo) == zeta(m + 1)
assert limit(harmonic(n, 2), n, oo) == pi**2/6
assert limit(harmonic(n, 3), n, oo) == -polygamma(2, 1)/2
def test_harmonic_limit():
n = Symbol('n')
m = Symbol('m', positive=True)
assert limit(harmonic(n, m + 1), n, oo) == zeta(m + 1)
assert limit(harmonic(n, 2), n, oo) == pi**2 / 6
assert limit(harmonic(n, 3), n, oo) == -polygamma(2, 1) / 2
def test_Limits_simple_4a():
assert limit((sqrt(x) - sqrt(a)) / (x - a), x,
a) == 1 / (2 * sqrt(a)) # Example 5
assert limit((sqrt(x) - 1) / (root(x, 3) - 1), x, 1) == Rational(3,
2) # 205
assert limit((sqrt(1 + x) - sqrt(1 - x)) / x, x, 0) == 1 # 207
assert limit(sqrt(x**2 - 5 * x + 6) - x, x, oo) == -Rational(5, 2) # 213
def test_exponential():
x = Symbol('x', extended_real=True)
assert limit((1 + x/n)**n, n, oo) == exp(x)
assert limit((1 + x/(2*n))**n, n, oo) == exp(x/2)
assert limit((1 + x/(2*n + 1))**n, n, oo) == exp(x/2)
assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2)
assert limit(1 + (1 + 1/x)**x, x, oo) == 1 + E
def test_issue_75():
assert limit(abs(log(x)), x, oo) == oo
assert limit(tan(abs(pi/2 + 1/x))/acosh(pi/2 + 1/x), x, oo) == -oo
assert limit(tan(abs(pi/2 - 1/x))/acosh(pi/2 - 1/x), x, oo) == +oo
assert limit(abs(log(2 + 1/x)) - log(2 + 1/x), x, oo) == 0
assert limit(abs(log(2 - 1/x)) - log(2 - 1/x), x, oo) == 0
def test_exponential():
x = Symbol('x', extended_real=True)
assert limit((1 + x/n)**n, n, oo) == exp(x)
assert limit((1 + x/(2*n))**n, n, oo) == exp(x/2)
assert limit((1 + x/(2*n + 1))**n, n, oo) == exp(x/2)
assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2)
assert limit(1 + (1 + 1/x)**x, x, oo) == 1 + S.Exp1
def test_Limits_simple_1():
assert limit((n + 1) * (n + 2) * (n + 3) / n**3, n, oo) == 1 # 172
assert limit(sqrt(n + 1) - sqrt(n), n, oo) == 0 # 179
assert limit((2 * x - 3) * (3 * x + 5) * (4 * x - 6) /
(3 * x**3 + x - 1), x, oo) == 8 # Example 1
assert limit(x / root(x**3 + 10, 3), x, oo) == 1 # Example 2
assert limit((x + 1)**2 / (x**2 + 1), x, oo) == 1 # 181
def test_gruntz_other():
assert limit(sqrt(log(x + 1)) - sqrt(log(x)), x, oo) == 0 # p12, 2.5
assert limit(((1 + 1/x)**y - 1)*x, x, oo) == y # p12, 2.6
assert limit(x**n/exp(x), x, oo) == 0 # p14, 2.9
assert limit((1 + 1/x)*x - 1/log(1 + 1/x),
x, oo) == Rational(1, 2) # p15, 2.10
assert limit((root(1 + 1/x, n) - 1)/(root(1 + 1/x, m) - 1),
x, oo) == m/n # p13, 2.7
def test_sympyissue_11526():
df = diff(1/(a*log((x - b)/(x - c))), x)
res = -1/(-a*c + a*b)
assert limit(df, x, oo) == res
assert limit(simplify(df), x, oo) == res
e = log((1/x - b)/(1/x - c))
assert e.as_leading_term(x) == x*(c - b)
def test_sympyissue_11526():
df = diff(1/(a*log((x - b)/(x - c))), x)
res = -1/(-a*c + a*b)
assert limit(df, x, oo) == res
assert limit(simplify(df), x, oo) == res
e = log((1/x - b)/(1/x - c))
assert e.as_leading_term(x) == x*(c - b)
def test_sympyissue_21029():
e = (sinh(x) + cosh(x) - 1)/x/4
ans0 = Rational(1, 4)
ansz = (exp(z) - 1)/z/4
assert limit(e, x, 0) == ans0
assert limit(e, x, z) == ansz
assert limit(ansz, z, 0) == ans0
def test_issue_55():
assert limit((x + exp(x))/(x - 1), x, -oo) == 1
assert limit((x*exp(x))/(exp(x) - 1), x, -oo) == 0 # issue sympy/sympy#2929
# issue sympy/sympy#22982
assert limit((log(E + 1/x) - 1)**(1 - sqrt(E + 1/x)), x, oo) == oo
assert limit((log(E + 1/x))**(1 - sqrt(E + 1/x)), x, oo) == 1
assert limit((log(E + 1/x) - 1)**-sqrt(E + 1/x), x, oo) == oo
def test_sympyissue_17431():
assert limit(((n + 1) + 1)/(((n + 1) + 2)*factorial(n + 1)) *
(n + 2)*factorial(n)/(n + 1), n, oo) == 0
assert limit((n + 2)**2*factorial(n)/((n + 1)*(n + 3)*factorial(n + 1)),
n, oo) == 0
# test from sympy/sympy#17434 (see also diofant/diofant#425):
y = symbols('y', integer=True, positive=True)
assert isinstance(limit(x*factorial(x)/factorial(x + y), x, oo), Limit)
def test_sympyissue_22220():
e1 = sqrt(30)*atan(sqrt(30)*tan(x/2)/6)/30
e2 = sqrt(30)*I*(-log(sqrt(2)*tan(x/2) - 2*sqrt(15)*I/5) +
+log(sqrt(2)*tan(x/2) + 2*sqrt(15)*I/5))/60
assert limit(e1, x, -pi) == -sqrt(30)*pi/60
assert limit(e2, x, -pi) == -sqrt(30)*pi/30
assert limit(e1, x, -pi, 1) == sqrt(30)*pi/60
assert limit(e2, x, -pi, 1) == 0
def test_sympyissue_5172():
r = Symbol('r', positive=True)
p = Symbol('p', positive=True)
m = Symbol('m', negative=True)
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.subs({c: m}), n, oo) == 1
assert limit(expr.subs({c: p}), n, oo) == (2**(p + 1) + r -
1)/(r + 1)**(p + 1)
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_sympyissue_5172():
r = Symbol('r', positive=True)
p = Symbol('p', positive=True)
m = Symbol('m', negative=True)
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.subs(c, m), n, oo) == 1
assert limit(expr.subs(c, p), n, oo).simplify() == \
(2**(p + 1) + r - 1)/(r + 1)**(p + 1)
def test_sympyissue_5172():
r = Symbol('r', positive=True)
p = Symbol('p', positive=True)
m = Symbol('m', negative=True)
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.subs({c: m}), n, oo) == 1
assert limit(expr.subs({c: p}), n, oo).simplify() == \
(2**(p + 1) + r - 1)/(r + 1)**(p + 1)
def test_sympyissue_8635():
n = Symbol('n', integer=True, positive=True)
k = 0
assert limit(x**n - x**(n - k), x, oo) == 0
k = 1
assert limit(x**n - x**(n - k), x, oo) == oo
k = 2
assert limit(x**n - x**(n - k), x, oo) == oo
k = 3
assert limit(x**n - x**(n - k), x, oo) == oo
def test_acosh():
assert limit(acosh(I*x), x, 0) == 2*log(1 + I) - log(2)
assert limit(acosh(I*x), x, 0, 1) == 2*log(1 - I) - log(2)
assert limit(acosh(-I*x), x, 0) == 2*log(1 - I) - log(2)
assert limit(acosh(-I*x), x, 0, 1) == 2*log(1 + I) - log(2)
pytest.raises(PoleError, lambda: limit(acosh(I*x), x, 0, Reals))
assert limit(acosh(x - I*x), x, 0) == 2*log(1 - I) - log(2)
assert limit(acosh(x - I*x), x, 0, 1) == 2*log(1 + I) - log(2)
pytest.raises(PoleError, lambda: limit(acosh(x - I*x), x, 0, Reals))
assert limit(acosh(I*x**2), x, 0, Reals) == 2*log(1 + I) - log(2)
assert limit(acosh(x**2 - I*x**2), x, 0, Reals) == 2*log(1 - I) - log(2)
def test_sympyissue_8635():
n = Symbol('n', integer=True, positive=True)
k = 0
assert limit(x**n - x**(n - k), x, oo) == 0
k = 1
assert limit(x**n - x**(n - k), x, oo) == oo
k = 2
assert limit(x**n - x**(n - k), x, oo) == oo
k = 3
assert limit(x**n - x**(n - k), x, oo) == oo
def test_sympyissue_16722():
n, z = symbols('n z')
assert isinstance(limit(binomial(n + z, n)*n**-z, n, oo), Limit)
z = symbols('z', positive=True)
assert limit(binomial(n + z, n)*n**-z, n, oo) == 1/gamma(z + 1)
n = symbols('n', positive=True, integer=True)
z = symbols('z', positive=True)
assert isinstance(limit(binomial(n + z, n)*n**-z, n, oo), Limit)
n, z = symbols('n z', positive=True, integer=True)
assert isinstance(limit(binomial(n + z, n)*n**-z, n, oo), Limit)
def test_sympyissue_19453():
beta = Symbol('beta', real=True, positive=True)
h = Symbol('h', real=True, positive=True)
m = Symbol('m', real=True, positive=True)
w = Symbol('omega', real=True, positive=True)
g = Symbol('g', real=True, positive=True)
q = 3*h**2*beta*g*exp(h*beta*w/2)
p = m**2*w**2
s = exp(h*beta*w) - 1
z = (-q/(4*p*s) - q/(2*p*s**2) -
q*(exp(h*beta*w) + 1)/(2*p*s**3) + exp(h*beta*w/2)/s)
e = -diff(log(z), beta)
assert limit(e - h*w/2, beta, oo) == 0
assert limit(e.simplify() - h*w/2, beta, oo) == 0
def test_floor_requires_robust_assumptions():
assert limit(floor(sin(x)), x, 0, "+") == 0
assert limit(floor(sin(x)), x, 0, "-") == -1
assert limit(floor(cos(x)), x, 0, "+") == 0
assert limit(floor(cos(x)), x, 0, "-") == 0
assert limit(floor(5 + sin(x)), x, 0, "+") == 5
assert limit(floor(5 + sin(x)), x, 0, "-") == 4
assert limit(floor(5 + cos(x)), x, 0, "+") == 5
assert limit(floor(5 + cos(x)), x, 0, "-") == 5
def test_Limits_simple_2():
assert limit(1000 * x / (x**2 - 1), x, oo) == 0 # 182
assert limit((x**2 - 5 * x + 1) / (3 * x + 7), x, oo) == oo # 183
assert limit((2 * x**2 - x + 3) / (x**3 - 8 * x + 5), x, oo) == 0 # 184
assert limit((2 * x**2 - 3 * x - 4) / sqrt(x**4 + 1), x, oo) == 2 # 186
assert limit((2 * x + 3) / (x + root3(x)), x, oo) == 2 # 187
assert limit(x**2 / (10 + x * sqrt(x)), x, oo) == oo # 188
assert limit(root3(x**2 + 1) / (x + 1), x, oo) == 0 # 189
assert limit(sqrt(x) / sqrt(x + sqrt(x + sqrt(x))), x, oo) == 1 # 190
def test_ceiling_requires_robust_assumptions():
assert limit(ceiling(sin(x)), x, 0, "+") == 1
assert limit(ceiling(sin(x)), x, 0, "-") == 0
assert limit(ceiling(cos(x)), x, 0, "+") == 1
assert limit(ceiling(cos(x)), x, 0, "-") == 1
assert limit(ceiling(5 + sin(x)), x, 0, "+") == 6
assert limit(ceiling(5 + sin(x)), x, 0, "-") == 5
assert limit(ceiling(5 + cos(x)), x, 0, "+") == 6
assert limit(ceiling(5 + cos(x)), x, 0, "-") == 6
def test_ceiling_requires_robust_assumptions():
assert limit(ceiling(sin(x)), x, 0, "+") == 1
assert limit(ceiling(sin(x)), x, 0, "-") == 0
assert limit(ceiling(cos(x)), x, 0, "+") == 1
assert limit(ceiling(cos(x)), x, 0, "-") == 1
assert limit(ceiling(5 + sin(x)), x, 0, "+") == 6
assert limit(ceiling(5 + sin(x)), x, 0, "-") == 5
assert limit(ceiling(5 + cos(x)), x, 0, "+") == 6
assert limit(ceiling(5 + cos(x)), x, 0, "-") == 6
def test_basic2():
assert limit(x**x, x, 0, dir="+") == 1
assert limit((exp(x) - 1)/x, x, 0) == 1
assert limit(1 + 1/x, x, oo) == 1
assert limit(-exp(1/x), x, oo) == -1
assert limit(x + exp(-x), x, oo) == oo
assert limit(x + exp(-x**2), x, oo) == oo
assert limit(x + exp(-exp(x)), x, oo) == oo
assert limit(13 + 1/x - exp(-x), x, oo) == 13
def test_basic2():
assert limit(x**x, x, 0, dir="+") == 1
assert limit((exp(x) - 1)/x, x, 0) == 1
assert limit(1 + 1/x, x, oo) == 1
assert limit(-exp(1/x), x, oo) == -1
assert limit(x + exp(-x), x, oo) == oo
assert limit(x + exp(-x**2), x, oo) == oo
assert limit(x + exp(-exp(x)), x, oo) == oo
assert limit(13 + 1/x - exp(-x), x, oo) == 13
def test_acoth():
assert limit(acoth(x), x, 0) == -I*pi/2
assert limit(acoth(x), x, 0, 1) == I*pi/2
pytest.raises(PoleError, lambda: limit(acoth(x), x, 0, Reals))
assert limit(acoth(-x), x, 0) == I*pi/2
assert limit(acoth(x**2), x, 0, Reals) == -I*pi/2
assert limit(acoth(I*x), x, 0) == -I*pi/2
assert limit(acoth(I*x), x, 0, 1) == I*pi/2
pytest.raises(PoleError, lambda: limit(acoth(x - I*x), x, 0, Reals))
def test_floor_requires_robust_assumptions():
assert limit(floor(sin(x)), x, 0, "+") == 0
assert limit(floor(sin(x)), x, 0, "-") == -1
assert limit(floor(cos(x)), x, 0, "+") == 0
assert limit(floor(cos(x)), x, 0, "-") == 0
assert limit(floor(5 + sin(x)), x, 0, "+") == 5
assert limit(floor(5 + sin(x)), x, 0, "-") == 4
assert limit(floor(5 + cos(x)), x, 0, "+") == 5
assert limit(floor(5 + cos(x)), x, 0, "-") == 5
def test_issue_74():
assert limit(sign(log(1 + 1/x)), x, oo) == +1
assert limit(sign(log(1 - 1/x)), x, oo) == -1
assert limit(sign(sin(+1/x)), x, oo) == +1
assert limit(sign(sin(-1/x)), x, oo) == -1
assert limit(sign(tan(+1/x)), x, oo) == +1
assert limit(sign(tan(-1/x)), x, oo) == -1
assert limit(sign(cos(pi/2 + 1/x)), x, oo) == -1
assert limit(sign(cos(pi/2 - 1/x)), x, oo) == +1
def test_sympyissue_11678():
p = Matrix([[1./2, 1./4, 1./4],
[1./2, 0, 1./2],
[1./4, 0, 3./4]])
e = (p**x).applyfunc(lambda i: limit(i, x, oo))
assert e == Matrix([[Float('0.36363636363636359', prec=15),
Float('0.090909090909090898', prec=15),
Float('0.54545454545454541', prec=15)]]*3)
def test_sympyissue_11678():
p = Matrix([[1./2, 1./4, 1./4],
[1./2, 0, 1./2],
[1./4, 0, 3./4]])
e = (p**x).applyfunc(lambda i: limit(i, x, oo))
assert e == Matrix([[Float('0.36363636363636359', dps=15),
Float('0.090909090909090898', dps=15),
Float('0.54545454545454541', dps=15)]]*3)
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_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 oo:
return 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_extended_real_line():
assert limit(x - oo, x, oo) == -oo
assert limit(oo - x, x, -oo) == oo
assert limit(x**2/(x - 5) - oo, x, oo) == -oo
assert limit(1/(x + sin(x)) - oo, x, 0) == -oo
assert limit(oo/x, x, oo) == oo
assert limit(x - oo + 1/x, x, oo) == -oo
assert limit(x - oo + 1/x, x, 0) == -oo
def test_sympyissue_4546():
# using list(...) so py.test can recalculate values
tests = list(itertools.product([cot, tan],
[-pi/2, 0, pi/2, pi, 3*pi/2],
['-', '+']))
results = (0, 0, -oo, oo, 0, 0, -oo, oo, 0, 0,
oo, -oo, 0, 0, oo, -oo, 0, 0, oo, -oo)
assert len(tests) == len(results)
for i, (args, res) in enumerate(zip(tests, results)):
f, l, d = args
eq = f(x)
assert limit(eq, x, l, dir=d) == res
def test_sympyissue_12769():
r, z, x = symbols('r,z,x', real=True)
a, b, s0, K, F0, s, T = symbols('a,b,s0,K,F0,s,T',
positive=True, real=True)
fx = (F0**b*K**b*r*s0 -
sqrt((F0**2*K**(2*b)*a**2*(b - 1) +
F0**(2*b)*K**2*a**2*(b - 1) +
F0**(2*b)*K**(2*b)*s0**2*(b - 1)*(b**2 - 2*b + 1) -
2*F0**(2*b)*K**(b + 1)*a*r*s0*(b**2 - 2*b + 1) +
2*F0**(b + 1)*K**(2*b)*a*r*s0*(b**2 - 2*b + 1) -
2*F0**(b + 1)*K**(b + 1)*a**2*(b - 1))/((b - 1)*(b**2 - 2*b + 1))))*(b*r - b - r + 1)
assert limit(fx, K, F0) == (F0**(2*b)*b*r**2*s0 - 2*F0**(2*b)*b*r*s0 +
F0**(2*b)*b*s0 - F0**(2*b)*r**2*s0 +
2*F0**(2*b)*r*s0 - F0**(2*b)*s0)
def test_sympyissue_5184():
assert limit(sin(x)/x, x, oo) == 0
assert limit(atan(x), x, oo) == pi/2
assert limit(gamma(x), x, oo) == oo
assert limit(cos(x)/x, x, oo) == 0
assert limit(gamma(x), x, Rational(1, 2)) == sqrt(pi)
r = Symbol('r', real=True)
assert limit(r*sin(1/r), r, 0) == 0
def test_sympyissue_5183():
# using list(...) so py.test can recalculate values
tests = list(itertools.product([x, -x],
[-1, 1],
[2, 3, Rational(1, 2), Rational(2, 3)],
['-', '+']))
results = (oo, oo, -oo, oo, -oo*I, oo, -oo*sign(cbrt(-1)), oo,
0, 0, 0, 0, 0, 0, 0, 0,
oo, oo, oo, -oo, oo, -oo*I, oo, -oo*sign(cbrt(-1)),
0, 0, 0, 0, 0, 0, 0, 0)
assert len(tests) == len(results)
for i, (args, res) in enumerate(zip(tests, results)):
y, s, e, d = args
eq = y**(s*e)
assert limit(eq, x, 0, dir=d) == res
def test_factorial():
f = factorial(x)
assert limit(f, x, oo) == oo
assert limit(x/f, x, oo) == 0
# see Stirling's approximation:
# https//en.wikipedia.org/wiki/Stirling's_approximation
assert limit(f/(sqrt(2*pi*x)*(x/E)**x), x, oo) == 1
assert limit(f, x, -oo) == factorial(-oo)
assert (limit(f, x, x**2) - factorial(x**2)).simplify() == 0
assert (limit(f, x, -x**2) - factorial(-x**2)).simplify() == 0
def test_abs():
assert limit(abs(x), x, 0) == 0
assert limit(abs(sin(x)), x, 0) == 0
assert limit(abs(cos(x)), x, 0) == 1
assert limit(abs(sin(x + 1)), x, 0) == sin(1)
# sympy/sympy#12398
assert limit(abs(log(n)/n**3), n, oo) == 0
expr = abs(log(n)/n**3)
expr2 = expr.subs({n: n + 1})
assert limit(n*(expr/expr2 - 1), n, oo) == 3
def test_function_series1():
"""Create our new "sin" function."""
class my_function(Function):
def fdiff(self, argindex=1):
return cos(self.args[0])
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg == 0:
return sympify(0)
# Test that the taylor series is correct
assert my_function(x).series(x, 0, 10) == sin(x).series(x, 0, 10)
assert limit(my_function(x)/x, x, 0) == 1
def test_ci():
m1 = exp_polar(I*pi)
m1_ = exp_polar(-I*pi)
pI = exp_polar(I*pi/2)
mI = exp_polar(-I*pi/2)
assert Ci(m1*x) == Ci(x) + I*pi
assert Ci(m1_*x) == Ci(x) - I*pi
assert Ci(pI*x) == Chi(x) + I*pi/2
assert Ci(mI*x) == Chi(x) - I*pi/2
assert Chi(m1*x) == Chi(x) + I*pi
assert Chi(m1_*x) == Chi(x) - I*pi
assert Chi(pI*x) == Ci(x) + I*pi/2
assert Chi(mI*x) == Ci(x) - I*pi/2
assert Ci(exp_polar(2*I*pi)*x) == Ci(x) + 2*I*pi
assert Chi(exp_polar(-2*I*pi)*x) == Chi(x) - 2*I*pi
assert Chi(exp_polar(2*I*pi)*x) == Chi(x) + 2*I*pi
assert Ci(exp_polar(-2*I*pi)*x) == Ci(x) - 2*I*pi
assert Ci(oo) == 0
assert Ci(-oo) == I*pi
assert Chi(oo) == oo
assert Chi(-oo) == oo
assert mytd(Ci(x), cos(x)/x, x)
assert mytd(Chi(x), cosh(x)/x, x)
assert mytn(Ci(x), Ci(x).rewrite(Ei),
Ei(x*exp_polar(-I*pi/2))/2 + Ei(x*exp_polar(I*pi/2))/2, x)
assert mytn(Chi(x), Chi(x).rewrite(Ei),
Ei(x)/2 + Ei(x*exp_polar(I*pi))/2 - I*pi/2, x)
assert tn_arg(Ci)
assert tn_arg(Chi)
assert Ci(x).nseries(x, n=4) == \
EulerGamma + log(x) - x**2/4 + x**4/96 + O(x**6)
assert Chi(x).nseries(x, n=4) == \
EulerGamma + log(x) + x**2/4 + x**4/96 + O(x**6)
assert limit(log(x) - Ci(2*x), x, 0) == -log(2) - EulerGamma
def test_sympyissue_14811():
assert limit(((1 + Rational(2, 3)**(x + 1))**2**x)/(2**Rational(4, 3)**(x - 1)), x, oo) == oo
def test_sympyissue_4329():
assert tan(x).series(x, pi/2, n=3).removeO() == \
-pi/6 + x/3 - 1/(x - pi/2)
assert cot(x).series(x, pi, n=3).removeO() == \
-x/3 + pi/3 + 1/(x - pi)
assert limit(tan(x)**tan(2*x), x, pi/4) == exp(-1)
def test_sympyissue_14590():
assert limit((n**3*((n + 1)/n)**n)/((n + 1)*(n + 2)*(n + 3)), n, oo) == E
def test_sympyissue_14793():
e = ((x + Rational(1, 2))*log(x) - x +
log(2*pi)/2 - log(factorial(x)) + 1/(12*x))*x**3
assert limit(e, x, oo) == Rational(1, 360)
def test_sympyissue_15055():
assert limit(n**3*((-n - 1)*sin(1/n) + (n + 2)*sin(1/(n + 1)))/(-n + 1), n, oo) == 1
def test_sympyissue_15146():
assert limit((n/2)*(-2*n**3 - 2*(n**3 - 1)*n**2*digamma(n**3 + 1) +
2*(n**3 - 1)*n**2*digamma(n**3 + n + 1) +
n + 3), n, oo) == Rational(1, 3)
def test_sympyissue_15323():
assert limit(((1 - 1/x)**x).diff(x), x, 1) == 1
def test_basic3():
assert limit(1/x, x, 0, dir="+") == oo
assert limit(1/x, x, 0, dir="-") == -oo
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
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)
