def _coeff_isneg(a):
"""Return True if the leading Number is negative.
Examples
========
>>> from sympy.core.function import _coeff_isneg
>>> from sympy import S, Symbol, oo, pi
>>> _coeff_isneg(-3*pi)
True
>>> _coeff_isneg(S(3))
False
>>> _coeff_isneg(-oo)
True
>>> _coeff_isneg(Symbol('n', negative=True)) # coeff is 1
False
"""
if a.is_Mul:
a = a.args[0]
return a.is_Number and a.is_negative
python类oo()的实例源码
def _eval_nseries(self, x, n, logx):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import limit, oo, powsimp
arg = self.args[0]
arg_series = arg._eval_nseries(x, n=n, logx=logx)
if arg_series.is_Order:
return 1 + arg_series
arg0 = limit(arg_series.removeO(), x, 0)
if arg0 in [-oo, oo]:
return self
t = Dummy("t")
exp_series = exp(t)._taylor(t, n)
o = exp_series.getO()
exp_series = exp_series.removeO()
r = exp(arg0)*exp_series.subs(t, arg_series - arg0)
r += C.Order(o.expr.subs(t, (arg_series - arg0)), x)
r = r.expand()
return powsimp(r, deep=True, combine='exp')
def _fourier_transform(f, x, k, a, b, name, simplify=True):
"""
Compute a general Fourier-type transform
F(k) = a int_-oo^oo exp(b*I*x*k) f(x) dx.
For suitable choice of a and b, this reduces to the standard Fourier
and inverse Fourier transforms.
"""
from sympy import exp, I, oo
F = integrate(a*f*exp(b*I*x*k), (x, -oo, oo))
if not F.has(Integral):
return _simplify(F, simplify), True
if not F.is_Piecewise:
raise IntegralTransformError(name, f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(name, f, 'integral in unexpected form')
return _simplify(F, simplify), cond
def _sine_cosine_transform(f, x, k, a, b, K, name, simplify=True):
"""
Compute a general sine or cosine-type transform
F(k) = a int_0^oo b*sin(x*k) f(x) dx.
F(k) = a int_0^oo b*cos(x*k) f(x) dx.
For suitable choice of a and b, this reduces to the standard sine/cosine
and inverse sine/cosine transforms.
"""
F = integrate(a*f*K(b*x*k), (x, 0, oo))
if not F.has(Integral):
return _simplify(F, simplify), True
if not F.is_Piecewise:
raise IntegralTransformError(name, f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(name, f, 'integral in unexpected form')
return _simplify(F, simplify), cond
def _hankel_transform(f, r, k, nu, name, simplify=True):
"""
Compute a general Hankel transform
.. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.
"""
from sympy import besselj, oo
F = integrate(f*besselj(nu, k*r)*r, (r, 0, oo))
if not F.has(Integral):
return _simplify(F, simplify), True
if not F.is_Piecewise:
raise IntegralTransformError(name, f, 'could not compute integral')
F, cond = F.args[0]
if F.has(Integral):
raise IntegralTransformError(name, f, 'integral in unexpected form')
return _simplify(F, simplify), cond
def test_solve_poly_rde_no_cancel():
# deg(b) large
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
oo, DE) == Poly(t + x, t)
# deg(b) small
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert solve_poly_rde(Poly(0, x), Poly(x/2 - S(1)/4, x), oo, DE) == \
Poly(x**2/4 - x/4, x)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
Poly(t + 1, t, x)
# deg(b) == deg(D) - 1
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert no_cancel_equal(Poly(1 - t, t),
Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
(Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))
def test_solve_poly_rde_cancel():
# exp
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert cancel_exp(Poly(2*x, t), Poly(2*x, t), 0, DE) == \
Poly(1, t)
assert cancel_exp(Poly(2*x, t), Poly((1 + 2*x)*t, t), 1, DE) == \
Poly(t, t)
# TODO: Add more exp tests, including tests that require is_deriv_in_field()
# primitive
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
# If the DecrementLevel context manager is working correctly, this shouldn't
# cause any problems with the further tests.
raises(NonElementaryIntegralException, lambda: cancel_primitive(Poly(1, t), Poly(t, t), oo, DE))
assert cancel_primitive(Poly(1, t), Poly(t + 1/x, t), 2, DE) == \
Poly(t, t)
assert cancel_primitive(Poly(4*x, t), Poly(4*x*t**2 + 2*t/x, t), 3, DE) == \
Poly(t**2, t)
# TODO: Add more primitive tests, including tests that require is_deriv_in_field()
def test_dmp_degree_in():
assert dmp_degree_in([[[]]], 0, 2) == -oo
assert dmp_degree_in([[[]]], 1, 2) == -oo
assert dmp_degree_in([[[]]], 2, 2) == -oo
assert dmp_degree_in([[[1]]], 0, 2) == 0
assert dmp_degree_in([[[1]]], 1, 2) == 0
assert dmp_degree_in([[[1]]], 2, 2) == 0
assert dmp_degree_in(f_4, 0, 2) == 9
assert dmp_degree_in(f_4, 1, 2) == 12
assert dmp_degree_in(f_4, 2, 2) == 8
assert dmp_degree_in(f_6, 0, 2) == 4
assert dmp_degree_in(f_6, 1, 2) == 4
assert dmp_degree_in(f_6, 2, 2) == 6
assert dmp_degree_in(f_6, 3, 3) == 3
raises(IndexError, lambda: dmp_degree_in([[1]], -5, 1))
def dup_degree(f):
"""
Return the leading degree of ``f`` in ``K[x]``.
Note that the degree of 0 is negative infinity (the SymPy object -oo).
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densebasic import dup_degree
>>> f = ZZ.map([1, 2, 0, 3])
>>> dup_degree(f)
3
"""
if not f:
return -oo
return len(f) - 1
def dmp_degree(f, u):
"""
Return the leading degree of ``f`` in ``x_0`` in ``K[X]``.
Note that the degree of 0 is negative infinity (the SymPy object -oo).
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densebasic import dmp_degree
>>> dmp_degree([[[]]], 2)
-oo
>>> f = ZZ.map([[2], [1, 2, 3]])
>>> dmp_degree(f, 1)
1
"""
if dmp_zero_p(f, u):
return -oo
else:
return len(f) - 1
def dmp_degree_list(f, u):
"""
Return a list of degrees of ``f`` in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densebasic import dmp_degree_list
>>> f = ZZ.map([[1], [1, 2, 3]])
>>> dmp_degree_list(f, 1)
(1, 2)
"""
degs = [-oo]*(u + 1)
_rec_degree_list(f, u, 0, degs)
return tuple(degs)
def main():
x = Symbol("x")
a = Symbol("a")
h = Symbol("h")
show( limit(sqrt(x**2 - 5*x + 6) - x, x, oo), -Rational(5)/2 )
show( limit(x*(sqrt(x**2 + 1) - x), x, oo), Rational(1)/2 )
show( limit(x - sqrt3(x**3 - 1), x, oo), Rational(0) )
show( limit(log(1 + exp(x))/x, x, -oo), Rational(0) )
show( limit(log(1 + exp(x))/x, x, oo), Rational(1) )
show( limit(sin(3*x)/x, x, 0), Rational(3) )
show( limit(sin(5*x)/sin(2*x), x, 0), Rational(5)/2 )
show( limit(((x - 1)/(x + 1))**x, x, oo), exp(-2))
def _coeff_isneg(a):
"""Return True if the leading Number is negative.
Examples
========
>>> from sympy.core.function import _coeff_isneg
>>> from sympy import S, Symbol, oo, pi
>>> _coeff_isneg(-3*pi)
True
>>> _coeff_isneg(S(3))
False
>>> _coeff_isneg(-oo)
True
>>> _coeff_isneg(Symbol('n', negative=True)) # coeff is 1
False
"""
if a.is_Mul:
a = a.args[0]
return a.is_Number and a.is_negative
def __new__(cls, *args):
if len(args) == 5:
args = [(args[0], args[1]), (args[2], args[3]), args[4]]
if len(args) != 3:
raise TypeError("args must be either as, as', bs, bs', z or "
"as, bs, z")
def tr(p):
if len(p) != 2:
raise TypeError("wrong argument")
return TupleArg(_prep_tuple(p[0]), _prep_tuple(p[1]))
arg0, arg1 = tr(args[0]), tr(args[1])
if Tuple(arg0, arg1).has(oo, zoo, -oo):
raise ValueError("G-function parameters must be finite")
if any((a - b).is_Integer and a - b > 0
for a in arg0[0] for b in arg1[0]):
raise ValueError("no parameter a1, ..., an may differ from "
"any b1, ..., bm by a positive integer")
# TODO should we check convergence conditions?
return Function.__new__(cls, arg0, arg1, args[2])
def _eval_nseries(self, x, n, logx):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import limit, oo, Order, powsimp
arg = self.args[0]
arg_series = arg._eval_nseries(x, n=n, logx=logx)
if arg_series.is_Order:
return 1 + arg_series
arg0 = limit(arg_series.removeO(), x, 0)
if arg0 in [-oo, oo]:
return self
t = Dummy("t")
exp_series = exp(t)._taylor(t, n)
o = exp_series.getO()
exp_series = exp_series.removeO()
r = exp(arg0)*exp_series.subs(t, arg_series - arg0)
r += Order(o.expr.subs(t, (arg_series - arg0)), x)
r = r.expand()
return powsimp(r, deep=True, combine='exp')
def test_order_at():
a = Poly(t**4, t)
b = Poly((t**2 + 1)**3*t, t)
c = Poly((t**2 + 1)**6*t, t)
d = Poly((t**2 + 1)**10*t**10, t)
e = Poly((t**2 + 1)**100*t**37, t)
p1 = Poly(t, t)
p2 = Poly(1 + t**2, t)
assert order_at(a, p1, t) == 4
assert order_at(b, p1, t) == 1
assert order_at(c, p1, t) == 1
assert order_at(d, p1, t) == 10
assert order_at(e, p1, t) == 37
assert order_at(a, p2, t) == 0
assert order_at(b, p2, t) == 3
assert order_at(c, p2, t) == 6
assert order_at(d, p1, t) == 10
assert order_at(e, p2, t) == 100
assert order_at(Poly(0, t), Poly(t, t), t) == oo
assert order_at_oo(Poly(t**2 - 1, t), Poly(t + 1), t) == \
order_at_oo(Poly(t - 1, t), Poly(1, t), t) == -1
assert order_at_oo(Poly(0, t), Poly(1, t), t) == oo
def test_solve_poly_rde_no_cancel():
# deg(b) large
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
oo, DE) == Poly(t + x, t)
# deg(b) small
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert solve_poly_rde(Poly(0, x), Poly(x/2 - S(1)/4, x), oo, DE) == \
Poly(x**2/4 - x/4, x)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
Poly(t + 1, t, x)
# deg(b) == deg(D) - 1
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert no_cancel_equal(Poly(1 - t, t),
Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
(Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))
def test_dmp_degree_in():
assert dmp_degree_in([[[]]], 0, 2) == -oo
assert dmp_degree_in([[[]]], 1, 2) == -oo
assert dmp_degree_in([[[]]], 2, 2) == -oo
assert dmp_degree_in([[[1]]], 0, 2) == 0
assert dmp_degree_in([[[1]]], 1, 2) == 0
assert dmp_degree_in([[[1]]], 2, 2) == 0
assert dmp_degree_in(f_4, 0, 2) == 9
assert dmp_degree_in(f_4, 1, 2) == 12
assert dmp_degree_in(f_4, 2, 2) == 8
assert dmp_degree_in(f_6, 0, 2) == 4
assert dmp_degree_in(f_6, 1, 2) == 4
assert dmp_degree_in(f_6, 2, 2) == 6
assert dmp_degree_in(f_6, 3, 3) == 3
raises(IndexError, lambda: dmp_degree_in([[1]], -5, 1))
def dup_degree(f):
"""
Return the leading degree of ``f`` in ``K[x]``.
Note that the degree of 0 is negative infinity (the SymPy object -oo).
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densebasic import dup_degree
>>> f = ZZ.map([1, 2, 0, 3])
>>> dup_degree(f)
3
"""
if not f:
return -oo
return len(f) - 1
def dmp_degree_list(f, u):
"""
Return a list of degrees of ``f`` in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densebasic import dmp_degree_list
>>> f = ZZ.map([[1], [1, 2, 3]])
>>> dmp_degree_list(f, 1)
(1, 2)
"""
degs = [-oo]*(u + 1)
_rec_degree_list(f, u, 0, degs)
return tuple(degs)
def test_integral0(n=4):
'''Make sure that the polynomials are orthonormal
'''
x = sympy.Symbol('x')
y = sympy.Symbol('y')
z = sympy.Symbol('z')
vals = numpy.concatenate(
orthopy.enr2.tree(n, numpy.array([x, y, z]), symbolic=True)
)
assert sympy.integrate(
vals[0] * sympy.exp(-x**2-y**2-z**2),
(x, -oo, +oo), (y, -oo, +oo), (z, -oo, +oo)
) == sympy.sqrt(sympy.sqrt(sympy.pi))**3
for val in vals[1:]:
assert sympy.integrate(
val * sympy.exp(-x**2-y**2-z**2),
(x, -oo, +oo), (y, -oo, +oo), (z, -oo, +oo)
) == 0
return
def test_integral0(n=4):
'''Make sure that the polynomials are orthonormal
'''
x = sympy.Symbol('x')
y = sympy.Symbol('y')
vals = numpy.concatenate(
orthopy.e2r2.tree(n, numpy.array([x, y]), symbolic=True)
)
assert sympy.integrate(
vals[0] * sympy.exp(-x**2-y**2), (x, -oo, +oo), (y, -oo, +oo)
) == sympy.sqrt(sympy.pi)
for val in vals[1:]:
assert sympy.integrate(
val * sympy.exp(-x**2-y**2), (x, -oo, +oo), (y, -oo, +oo)
) == 0
return
def timeit_limit_1x():
limit(1/x, x, oo)
def _combine_inverse(lhs, rhs):
"""
Returns lhs - rhs, but treats arguments like symbols, so things like
oo - oo return 0, instead of a nan.
"""
from sympy import oo, I, expand_mul
if lhs == oo and rhs == oo or lhs == oo*I and rhs == oo*I:
return S.Zero
return expand_mul(lhs - rhs)
def test_issue_4124():
from sympy import oo
assert expand_complex(I*oo) == oo*I
def test_sympy__stats__crv__SingleContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain
assert _test_args(SingleContinuousDomain(x, Interval(-oo, oo)))
def test_sympy__stats__crv__ProductContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain, ProductContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
E = SingleContinuousDomain(y, Interval(0, oo))
assert _test_args(ProductContinuousDomain(D, E))
def test_sympy__stats__crv__ConditionalContinuousDomain():
from sympy.stats.crv import (SingleContinuousDomain,
ConditionalContinuousDomain)
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ConditionalContinuousDomain(D, x > 0))
def test_sympy__stats__crv__ContinuousPSpace():
from sympy.stats.crv import ContinuousPSpace, SingleContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ContinuousPSpace(D, nd))
def test_sympy__stats__rv__ProductDomain():
from sympy.stats.rv import ProductDomain, SingleDomain
D = SingleDomain(x, Interval(-oo, oo))
E = SingleDomain(y, Interval(0, oo))
assert _test_args(ProductDomain(D, E))
