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_airyai():
z = Symbol('z', 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**3/9)/(3*gamma(Rational(1, 3))) +
3**Rational(1, 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*(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 expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
# TODO: support discrete sets with non integer stepsizes
if evaluate:
try:
p = poly(expr, var)
t = Dummy('t', real=True)
mgf = self.moment_generating_function(t)
deg = p.degree()
taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
result = 0
for k in range(deg + 1):
result += p.coeff_monomial(var**k) * taylor.coeff_monomial(
t**k) * factorial(k)
return result
except PolynomialError:
return summation(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
else:
return Sum(expr * self.pdf(var), (var, self.set.inf, self.set.sup),
**kwargs)
def test_airybiprime():
z = Symbol('z', real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airybiprime(z), airybiprime)
assert airybiprime(0) == 3**Rational(1, 6)/gamma(Rational(1, 3))
assert airybiprime(oo) is oo
assert airybiprime(-oo) == 0
assert diff(airybiprime(z), z) == z*airybi(z)
assert series(airybiprime(z), z, 0, 3) == (
3**Rational(1, 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))) +
3**Rational(1, 6)*hyper((), (Rational(1, 3),), z**3/9)/gamma(Rational(1, 3)))
assert isinstance(airybiprime(z).rewrite(besselj), airybiprime)
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 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 expand_func(airybiprime(2*(3*z**5)**Rational(1, 3))) == (
sqrt(3)*(z**Rational(5, 3)/(z**5)**Rational(1, 3) - 1)*airyaiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2 +
(z**Rational(5, 3)/(z**5)**Rational(1, 3) + 1)*airybiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
def test_issue_14384():
x, a = symbols('x a')
assert series(x**a, x) == x**a
assert series(x**(-2*a), x) == x**(-2*a)
assert series(exp(a*log(x)), x) == exp(a*log(x))
assert series(x**I, x) == x**I
assert series(x**(I + 1), x) == x**(1 + I)
assert series(exp(I*log(x)), x) == exp(I*log(x))
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 expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
if evaluate:
try:
p = poly(expr, var)
if p.is_zero:
return S.Zero
t = Dummy('t', real=True)
mgf = self._moment_generating_function(t)
if mgf is None:
return integrate(expr * self.pdf(var), (var, self.set), **kwargs)
deg = p.degree()
taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
result = 0
for k in range(deg+1):
result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k)
return result
except PolynomialError:
return integrate(expr * self.pdf(var), (var, self.set), **kwargs)
else:
return Integral(expr * self.pdf(var), (var, self.set), **kwargs)
def test_sin():
e1 = sin(x).series(x, 0)
e2 = series(sin(x), x, 0)
assert e1 == e2
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_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_exp2():
e1 = exp(cos(x)).series(x, 0)
e2 = series(exp(cos(x)), x, 0)
assert e1 == e2
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_exp():
e1 = exp(x).series(x, 0)
e2 = series(exp(x), x, 0)
assert e1 == e2
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_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_cos():
e1 = cos(x).series(x, 0)
e2 = series(cos(x), x, 0)
assert e1 == e2
def test_issue_7203():
assert series(cos(x), x, pi, 3) == \
-1 + (x - pi)**2/2 + O((x - pi)**3, (x, pi))
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_8805():
assert series(1, n=8) == 1
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)
