def _generic_mul(q1, q2):
q1 = sympify(q1)
q2 = sympify(q2)
# None is a Quaternion:
if not isinstance(q1, Quaternion) and not isinstance(q2, Quaternion):
return q1 * q2
# If q1 is a number or a sympy expression instead of a quaternion
if not isinstance(q1, Quaternion):
if q2.real_field:
if q1.is_complex:
return q2 * Quaternion(re(q1), im(q1), 0, 0)
else:
return Mul(q1, q2)
else:
return Quaternion(q1 * q2.a, q1 * q2.b, q1 * q2.c, q1 * q2.d)
# If q2 is a number or a sympy expression instead of a quaternion
if not isinstance(q2, Quaternion):
if q1.real_field:
if q2.is_complex:
return q1 * Quaternion(re(q2), im(q2), 0, 0)
else:
return Mul(q1, q2)
else:
return Quaternion(q2 * q1.a, q2 * q1.b, q2 * q1.c, q2 * q1.d)
return Quaternion(-q1.b*q2.b - q1.c*q2.c - q1.d*q2.d + q1.a*q2.a,
q1.b*q2.a + q1.c*q2.d - q1.d*q2.c + q1.a*q2.b,
-q1.b*q2.d + q1.c*q2.a + q1.d*q2.b + q1.a*q2.c,
q1.b*q2.c - q1.c*q2.b + q1.d*q2.a + q1.a * q2.d)
def velocity_field(psi): #takes a symbolic function and returns two lambda functions
#to evaluate the derivatives in both x and y.
global w
if velocity_components:
u = lambdify((x,y), eval(x_velocity), modules='numpy')
v = lambdify((x,y), eval(y_velocity), modules='numpy')
else:
if is_complex_potential:
print "Complex potential, w(z) given"
#define u, v symbolically as the imaginary part of the derivatives
u = lambdify((x, y), sympy.im(psi.diff(y)), modules='numpy')
v = lambdify((x, y), -sympy.im(psi.diff(x)), modules='numpy')
else:
#define u,v as the derivatives
print "Stream function, psi given"
u = sympy.lambdify((x, y), psi.diff(y), 'numpy')
v = sympy.lambdify((x, y), -psi.diff(x), 'numpy')
if (branch_cuts): # If it's indicated that there are branch cuts in the mapping,
# then we need to return vectorized numpy functions to evaluate
# everything numerically, instead of symbolically
# This of course results in a SIGNIFICANT time increase
# (I don't know how to handle more than the primitive root
# (symbolically in Sympy
return np.vectorize(u),np.vectorize(v)
else:
# If there are no branch cuts, then return the symbolic lambda functions (MUCH faster)
return u,v
def test_as_real_imag():
n = pi**1000
# the special code for working out the real
# and complex parts of a power with Integer exponent
# should not run if there is no imaginary part, hence
# this should not hang
assert n.as_real_imag() == (n, 0)
# issue 6261
x = Symbol('x')
assert sqrt(x).as_real_imag() == \
((re(x)**2 + im(x)**2)**(S(1)/4)*cos(atan2(im(x), re(x))/2),
(re(x)**2 + im(x)**2)**(S(1)/4)*sin(atan2(im(x), re(x))/2))
# issue 3853
a, b = symbols('a,b', real=True)
assert ((1 + sqrt(a + b*I))/2).as_real_imag() == \
(
(a**2 + b**2)**Rational(
1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2),
(a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2)
assert sqrt(a**2).as_real_imag() == (sqrt(a**2), 0)
i = symbols('i', imaginary=True)
assert sqrt(i**2).as_real_imag() == (0, abs(i))
def test_f_expand_complex():
x = Symbol("x", real=True)
assert f(x).expand(complex=True) == I * im(f(x)) + re(f(x))
assert exp(x).expand(complex=True) == exp(x)
assert exp(I * x).expand(complex=True) == cos(x) + I * sin(x)
assert exp(z).expand(complex=True) == cos(im(z)) * exp(re(z)) + I * sin(im(z)) * exp(re(z))
def test_re():
x, y = symbols('x,y')
a, b = symbols('a,b', real=True)
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
assert re(nan) == nan
assert re(oo) == oo
assert re(-oo) == -oo
assert re(0) == 0
assert re(1) == 1
assert re(-1) == -1
assert re(E) == E
assert re(-E) == -E
assert re(x) == re(x)
assert re(x*I) == -im(x)
assert re(r*I) == 0
assert re(r) == r
assert re(i*I) == I * i
assert re(i) == 0
assert re(x + y) == re(x + y)
assert re(x + r) == re(x) + r
assert re(re(x)) == re(x)
assert re(2 + I) == 2
assert re(x + I) == re(x)
assert re(x + y*I) == re(x) - im(y)
assert re(x + r*I) == re(x)
assert re(log(2*I)) == log(2)
assert re((2 + I)**2).expand(complex=True) == 3
assert re(conjugate(x)) == re(x)
assert conjugate(re(x)) == re(x)
assert re(x).as_real_imag() == (re(x), 0)
assert re(i*r*x).diff(r) == re(i*x)
assert re(i*r*x).diff(i) == I*r*im(x)
assert re(
sqrt(a + b*I)) == (a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)
assert re(a * (2 + b*I)) == 2*a
assert re((1 + sqrt(a + b*I))/2) == \
(a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2)
assert re(x).rewrite(im) == x - im(x)
assert (x + re(y)).rewrite(re, im) == x + y - im(y)
def test_reduce_inequalities_multivariate():
x = Symbol('x')
y = Symbol('y')
assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == \
And(Eq(im(x), 0), Eq(im(y), 0), Or(And(Le(1, re(x)), Lt(re(x), oo)),
And(Le(re(x), -1), Lt(-oo, re(x)))),
Or(And(Le(1, re(y)), Lt(re(y), oo)), And(Le(re(y), -1), Lt(-oo, re(y)))))
def test_derivatives_issue1658():
x = Symbol('x')
f = Function('f')
assert re(f(x)).diff(x) == re(f(x).diff(x))
assert im(f(x)).diff(x) == im(f(x).diff(x))
x = Symbol('x', real=True)
assert Abs(f(x)).diff(x).subs(f(x), 1+I*x).doit() == x/sqrt(1 + x**2)
assert arg(f(x)).diff(x).subs(f(x), 1+I*x**2).doit() == 2*x/(1+x**4)
def test_f_expand_complex():
f = Function('f')
x = Symbol('x', real=True)
z = Symbol('z')
assert f(x).expand(complex=True) == I*im(f(x)) + re(f(x))
assert exp(x).expand(complex=True) == exp(x)
assert exp(I*x).expand(complex=True) == cos(x) + I*sin(x)
assert exp(z).expand(complex=True) == cos(im(z))*exp(re(z)) + \
I*sin(im(z))*exp(re(z))
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1)+exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'
beta = Function('beta')
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2,inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2,inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2),inv_trig_style="power",fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2,k)) == r"{\binom{2}{k}}"
assert latex(FallingFactorial(3,k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3,k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Abs(x)) == r"\lvert{x}\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x+y)) == r"\Re {\left (x + y \right )}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x,y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta{\left (x \right )}'
def _eval_evalf(self, prec):
""" Careful! any evalf of polar numbers is flaky """
from sympy import im, pi, re
i = im(self.args[0])
if i <= -pi or i > pi:
return self # cannot evalf for this argument
res = exp(self.args[0])._eval_evalf(prec)
if i > 0 and im(res) < 0:
# i ~ pi, but exp(I*i) evaluated to argument slightly bigger than pi
return re(res)
return res
def test_map_coeffs():
p = Poly(x**2 + 2*x*y, x, y)
q = p.map_coeffs(lambda c: 2*c)
assert q.as_basic() == 2*x**2 + 4*x*y
p = Poly(u*x**2 + v*x*y, x, y)
q = p.map_coeffs(expand, complex=True)
assert q.as_basic() == x**2*(I*im(u) + re(u)) + x*y*(I*im(v) + re(v))
raises(PolynomialError, "p.map_coeffs(lambda c: x*c)")
def test_derivatives_issue_4757():
x = Symbol('x', real=True)
y = Symbol('y', imaginary=True)
f = Function('f')
assert re(f(x)).diff(x) == re(f(x).diff(x))
assert im(f(x)).diff(x) == im(f(x).diff(x))
assert re(f(y)).diff(y) == -I*im(f(y).diff(y))
assert im(f(y)).diff(y) == -I*re(f(y).diff(y))
assert Abs(f(x)).diff(x).subs(f(x), 1 + I*x).doit() == x/sqrt(1 + x**2)
assert arg(f(x)).diff(x).subs(f(x), 1 + I*x**2).doit() == 2*x/(1 + x**4)
assert Abs(f(y)).diff(y).subs(f(y), 1 + y).doit() == -y/sqrt(1 - y**2)
assert arg(f(y)).diff(y).subs(f(y), I + y**2).doit() == 2*y/(1 + y**4)
def test_as_real_imag():
n = pi**1000
# the special code for working out the real
# and complex parts of a power with Integer exponent
# should not run if there is no imaginary part, hence
# this should not hang
assert n.as_real_imag() == (n, 0)
# issue 3162
x = Symbol('x')
assert sqrt(x).as_real_imag() == \
((re(x)**2 + im(x)**2)**(S(1)/4)*cos(atan2(im(x), re(x))/2), \
(re(x)**2 + im(x)**2)**(S(1)/4)*sin(atan2(im(x), re(x))/2))
def eval(cls, arg):
from sympy import im
if arg.is_integer or arg.is_finite is False:
return arg
if arg.is_imaginary or (S.ImaginaryUnit*arg).is_real:
i = im(arg)
if not i.has(S.ImaginaryUnit):
return cls(i)*S.ImaginaryUnit
return cls(arg, evaluate=False)
v = cls._eval_number(arg)
if v is not None:
return v
# Integral, numerical, symbolic part
ipart = npart = spart = S.Zero
# Extract integral (or complex integral) terms
terms = Add.make_args(arg)
for t in terms:
if t.is_integer or (t.is_imaginary and im(t).is_integer):
ipart += t
elif t.has(Symbol):
spart += t
else:
npart += t
if not (npart or spart):
return ipart
# Evaluate npart numerically if independent of spart
if npart and (
not spart or
npart.is_real and (spart.is_imaginary or (S.ImaginaryUnit*spart).is_real) or
npart.is_imaginary and spart.is_real):
try:
r, i = get_integer_part(
npart, cls._dir, {}, return_ints=True)
ipart += Integer(r) + Integer(i)*S.ImaginaryUnit
npart = S.Zero
except (PrecisionExhausted, NotImplementedError):
pass
spart += npart
if not spart:
return ipart
elif spart.is_imaginary or (S.ImaginaryUnit*spart).is_real:
return ipart + cls(im(spart), evaluate=False)*S.ImaginaryUnit
else:
return ipart + cls(spart, evaluate=False)
def test_zero_assumptions():
nr = Symbol('nonreal', real=False)
ni = Symbol('nonimaginary', imaginary=False)
# imaginary implies not zero
nzni = Symbol('nonzerononimaginary', zero=False, imaginary=False)
assert re(nr).is_zero is None
assert im(nr).is_zero is False
assert re(ni).is_zero is None
assert im(ni).is_zero is None
assert re(nzni).is_zero is False
assert im(nzni).is_zero is None
def test_diff_no_eval_derivative():
class My(Expr):
def __new__(cls, x):
return Expr.__new__(cls, x)
x, y = symbols('x y')
# My doesn't have its own _eval_derivative method
assert My(x).diff(x).func is Derivative
assert My(x).diff(x, 3).func is Derivative
assert re(x).diff(x, 2) == Derivative(re(x), (x, 2)) # issue 15518
assert diff(NDimArray([re(x), im(x)]), (x, 2)) == NDimArray(
[Derivative(re(x), (x, 2)), Derivative(im(x), (x, 2))])
# it doesn't have y so it shouldn't need a method for this case
assert My(x).diff(y) == 0
def test_real_imag():
x, y, z = symbols('x, y, z')
X, Y, Z = symbols('X, Y, Z', commutative=False)
a = Symbol('a', real=True)
assert (2*a*x).as_real_imag() == (2*a*re(x), 2*a*im(x))
# issue 2296:
assert (x*x.conjugate()).as_real_imag() == (Abs(x)**2, 0)
assert im(x*x.conjugate()) == 0
assert im(x*y.conjugate()*z*y) == im(x*z)*Abs(y)**2
assert im(x*y.conjugate()*x*y) == im(x**2)*Abs(y)**2
assert im(Z*y.conjugate()*X*y) == im(Z*X)*Abs(y)**2
assert im(X*X.conjugate()) == im(X*X.conjugate(), evaluate=False)
assert (sin(x)*sin(x).conjugate()).as_real_imag() == \
(Abs(sin(x))**2, 0)
def test_re():
x, y = symbols('x,y')
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
assert re(nan) == nan
assert re(oo) == oo
assert re(-oo) == -oo
assert re(0) == 0
assert re(1) == 1
assert re(-1) == -1
assert re(E) == E
assert re(-E) == -E
assert re(x) == re(x)
assert re(x*I) == -im(x)
assert re(r*I) == 0
assert re(r) == r
assert re(i*I) == I * i
assert re(i) == 0
assert re(x + y) == re(x + y)
assert re(x + r) == re(x) + r
assert re(re(x)) == re(x)
assert re(2 + I) == 2
assert re(x + I) == re(x)
assert re(x + y*I) == re(x) - im(y)
assert re(x + r*I) == re(x)
assert re(log(2*I)) == log(2)
assert re((2+I)**2).expand(complex=True) == 3
assert re(conjugate(x)) == re(x)
assert conjugate(re(x)) == re(x)
assert re(x).as_real_imag() == (re(x), 0)
assert re(i*r*x).diff(r) == re(i*x)
assert re(i*r*x).diff(i) == -I * im(r*x)
def plot_initial_monodromy_path(self, i, Npts=4, eps=0.1):
"""
Plots the initial monodromy path. Used for testing and
debugging purposes.
"""
# plot the monodromy graph
self.show_paths()
fig = plt.gcf()
ax = fig.gca()
# compute the path points
path_points = self._initial_monodromy_path(i, Npts=Npts)
N = len(path_points)
# plot forward path
topos = lambda pp: (sympy.re(pp), sympy.im(pp))
for n in xrange(N/2):
pos = topos(path_points[n])
ax.text(pos[0], pos[1], '%d'%n, size='x-small',
verticalalignment='top')
# plot return path
for n in xrange(N/2,N):
pos = topos(path_points[n])
ax.text(pos[0], pos[1], '%d'%n, size='x-small',
verticalalignment='bottom')
ax.axis('tight')
fig.show()
def test_evalc():
x = Symbol("x", real=True)
y = Symbol("y", real=True)
z = Symbol("z")
assert ((x+I*y)**2).expand(complex=True) == x**2+2*I*x*y - y**2
assert expand_complex(z**(2*I)) == I*im(z**(2*I)) + re(z**(2*I))
assert exp(I*x) != cos(x)+I*sin(x)
assert exp(I*x).expand(complex=True) == cos(x)+I*sin(x)
assert exp(I*x+y).expand(complex=True) == exp(y)*cos(x)+I*sin(x)*exp(y)
assert sin(I*x).expand(complex=True) == I * sinh(x)
assert sin(x+I*y).expand(complex=True) == sin(x)*cosh(y) + \
I * sinh(y) * cos(x)
assert cos(I*x).expand(complex=True) == cosh(x)
assert cos(x+I*y).expand(complex=True) == cos(x)*cosh(y) - \
I * sinh(y) * sin(x)
assert tan(I*x).expand(complex=True) == tanh(x) * I
assert tan(x+I*y).expand(complex=True) == \
((sin(x)*cos(x) + I*cosh(y)*sinh(y)) / (cos(x)**2 + sinh(y)**2)).expand()
assert sinh(I*x).expand(complex=True) == I * sin(x)
assert sinh(x+I*y).expand(complex=True) == sinh(x)*cos(y) + \
I * sin(y) * cosh(x)
assert cosh(I*x).expand(complex=True) == cos(x)
assert cosh(x+I*y).expand(complex=True) == cosh(x)*cos(y) + \
I * sin(y) * sinh(x)
assert tanh(I*x).expand(complex=True) == tan(x) * I
assert tanh(x+I*y).expand(complex=True) == \
((sinh(x)*cosh(x) + I*cos(y)*sin(y)) / (sinh(x)**2 + cos(y)**2)).expand()
def test_imag_real():
f_re = lambdify([z], sympy.re(z))
val = 3+2j
assert f_re(val) == val.real
f_im = lambdify([z], sympy.im(z)) # see #15400
assert f_im(val) == val.imag
def test_catalan():
n = Symbol('n', integer=True)
m = Symbol('n', integer=True, positive=True)
catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
for i, c in enumerate(catalans):
assert catalan(i) == c
assert catalan(n).rewrite(factorial).subs(n, i) == c
assert catalan(n).rewrite(Product).subs(n, i).doit() == c
assert catalan(x) == catalan(x)
assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8/(3*pi)
assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\
8 / (3 * pi)
assert catalan(3*x).rewrite(gamma) == 4**(
3*x)*gamma(3*x + Rational(1, 2))/(sqrt(pi)*gamma(3*x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1)
assert catalan(n).rewrite(factorial) == factorial(2*n) / (factorial(n + 1)
* factorial(n))
assert isinstance(catalan(n).rewrite(Product), catalan)
assert isinstance(catalan(m).rewrite(Product), Product)
assert diff(catalan(x), x) == (polygamma(
0, x + Rational(1, 2)) - polygamma(0, x + 2) + log(4))*catalan(x)
assert catalan(x).evalf() == catalan(x)
c = catalan(S.Half).evalf()
assert str(c) == '0.848826363156775'
c = catalan(I).evalf(3)
assert str((re(c), im(c))) == '(0.398, -0.0209)'
def _eval_evalf(self, prec):
""" Careful! any evalf of polar numbers is flaky """
from sympy import im, pi
i = im(self.args[0])
if i <= -pi or i > pi:
return self # cannot evalf for this argument
return exp(self.args[0])._eval_evalf(prec)
def get_integer_part(expr, no, options, return_ints=False):
"""
With no = 1, computes ceiling(expr)
With no = -1, computes floor(expr)
Note: this function either gives the exact result or signals failure.
"""
import sympy
# The expression is likely less than 2^30 or so
assumed_size = 30
ire, iim, ire_acc, iim_acc = evalf(expr, assumed_size, options)
# We now know the size, so we can calculate how much extra precision
# (if any) is needed to get within the nearest integer
if ire and iim:
gap = max(fastlog(ire) - ire_acc, fastlog(iim) - iim_acc)
elif ire:
gap = fastlog(ire) - ire_acc
elif iim:
gap = fastlog(iim) - iim_acc
else:
# ... or maybe the expression was exactly zero
return None, None, None, None
margin = 10
if gap >= -margin:
ire, iim, ire_acc, iim_acc = \
evalf(expr, margin + assumed_size + gap, options)
# We can now easily find the nearest integer, but to find floor/ceil, we
# must also calculate whether the difference to the nearest integer is
# positive or negative (which may fail if very close).
def calc_part(expr, nexpr):
from sympy import Add
nint = int(to_int(nexpr, rnd))
n, c, p, b = nexpr
if (c != 1 and p != 0) or p < 0:
expr = Add(expr, -nint, evaluate=False)
x, _, x_acc, _ = evalf(expr, 10, options)
try:
check_target(expr, (x, None, x_acc, None), 3)
except PrecisionExhausted:
if not expr.equals(0):
raise PrecisionExhausted
x = fzero
nint += int(no*(mpf_cmp(x or fzero, fzero) == no))
nint = from_int(nint)
return nint, fastlog(nint) + 10
re, im, re_acc, im_acc = None, None, None, None
if ire:
re, re_acc = calc_part(sympy.re(expr, evaluate=False), ire)
if iim:
im, im_acc = calc_part(sympy.im(expr, evaluate=False), iim)
if return_ints:
return int(to_int(re or fzero)), int(to_int(im or fzero))
return re, im, re_acc, im_acc
def eval(cls, arg):
from sympy import AccumBounds, im
def _eval(arg):
if arg is S.Infinity or arg is S.NegativeInfinity:
return AccumBounds(0, 1)
if arg.is_integer:
return S.Zero
if arg.is_number:
if arg is S.NaN:
return S.NaN
elif arg is S.ComplexInfinity:
return None
else:
return arg - floor(arg)
return cls(arg, evaluate=False)
terms = Add.make_args(arg)
real, imag = S.Zero, S.Zero
for t in terms:
# Two checks are needed for complex arguments
# see issue-7649 for details
if t.is_imaginary or (S.ImaginaryUnit*t).is_real:
i = im(t)
if not i.has(S.ImaginaryUnit):
imag += i
else:
real += t
else:
real += t
real = _eval(real)
imag = _eval(imag)
return real + S.ImaginaryUnit*imag
def real_root(arg, n=None):
"""Return the real nth-root of arg if possible. If n is omitted then
all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this
will only create a real root of a principle root -- the presence of
other factors may cause the result to not be real.
Examples
========
>>> from sympy import root, real_root, Rational
>>> from sympy.abc import x, n
>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*(-1)**(1/3)
>>> real_root(_)
-2
If one creates a non-principle root and applies real_root, the
result will not be real (so use with caution):
>>> root(-8, 3, 2)
-2*(-1)**(2/3)
>>> real_root(_)
-2*(-1)**(2/3)
See Also
========
sympy.polys.rootoftools.RootOf
sympy.core.power.integer_nthroot
root, sqrt
"""
from sympy import im, Piecewise
if n is not None:
try:
n = as_int(n)
arg = sympify(arg)
if arg.is_positive or arg.is_negative:
rv = root(arg, n)
else:
raise ValueError
except ValueError:
return root(arg, n)*Piecewise(
(S.One, ~Equality(im(arg), 0)),
(Pow(S.NegativeOne, S.One/n)**(2*floor(n/2)), And(
Equality(n % 2, 1),
arg < 0)),
(S.One, True))
else:
rv = sympify(arg)
n1pow = Transform(lambda x: -(-x.base)**x.exp,
lambda x:
x.is_Pow and
x.base.is_negative and
x.exp.is_Rational and
x.exp.p == 1 and x.exp.q % 2)
return rv.xreplace(n1pow)
def test_reduce_poly_inequalities_complex_relational():
cond = Eq(im(x), 0)
assert reduce_poly_inequalities(
[[Eq(x**2, 0)]], x, relational=True) == And(Eq(re(x), 0), cond)
assert reduce_poly_inequalities(
[[Le(x**2, 0)]], x, relational=True) == And(Eq(re(x), 0), cond)
assert reduce_poly_inequalities(
[[Lt(x**2, 0)]], x, relational=True) is False
assert reduce_poly_inequalities(
[[Ge(x**2, 0)]], x, relational=True) == cond
assert reduce_poly_inequalities([[Gt(x**2, 0)]], x, relational=True) == And(Or(Lt(re(x), 0), Lt(0, re(x))), cond)
assert reduce_poly_inequalities([[Ne(x**2, 0)]], x, relational=True) == And(Or(Lt(re(x), 0), Lt(0, re(x))), cond)
assert reduce_poly_inequalities([[Eq(x**2, 1)]], x, relational=True) == And(Or(Eq(re(x), -1), Eq(re(x), 1)), cond)
assert reduce_poly_inequalities([[Le(x**2, 1)]], x, relational=True) == And(And(Le(-1, re(x)), Le(re(x), 1)), cond)
assert reduce_poly_inequalities([[Lt(x**2, 1)]], x, relational=True) == And(And(Lt(-1, re(x)), Lt(re(x), 1)), cond)
assert reduce_poly_inequalities([[Ge(x**2, 1)]], x, relational=True) == And(Or(Le(re(x), -1), Le(1, re(x))), cond)
assert reduce_poly_inequalities([[Gt(x**2, 1)]], x, relational=True) == And(Or(Lt(re(x), -1), Lt(1, re(x))), cond)
assert reduce_poly_inequalities([[Ne(x**2, 1)]], x, relational=True) == And(Or(Lt(re(x), -1), And(Lt(-1, re(x)), Lt(re(x), 1)), Lt(1, re(x))), cond)
assert reduce_poly_inequalities([[Eq(x**2, 1.0)]], x, relational=True).evalf() == And(Or(Eq(re(x), -1.0), Eq(re(x), 1.0)), cond)
assert reduce_poly_inequalities([[Le(x**2, 1.0)]], x, relational=True) == And(And(Le(-1.0, re(x)), Le(re(x), 1.0)), cond)
assert reduce_poly_inequalities([[Lt(x**2, 1.0)]], x, relational=True) == And(And(Lt(-1.0, re(x)), Lt(re(x), 1.0)), cond)
assert reduce_poly_inequalities([[Ge(x**2, 1.0)]], x, relational=True) == And(Or(Le(re(x), -1.0), Le(1.0, re(x))), cond)
assert reduce_poly_inequalities([[Gt(x**2, 1.0)]], x, relational=True) == And(Or(Lt(re(x), -1.0), Lt(1.0, re(x))), cond)
assert reduce_poly_inequalities([[Ne(x**2, 1.0)]], x, relational=True) == And(Or(Lt(re(x), -1.0), And(Lt(-1.0, re(x)), Lt(re(x), 1.0)), Lt(1.0, re(x))), cond)
def attr_draw( z, func, scale, displays ):
niter = 12
recurse = z - func / sympy.diff( func, z )
for x in range( -width/2, width/2 ):
for y in range( -height/2, height/2 ):
val = y*scale*sympy.I + x*scale
for i in range( niter ):
val = recurse.subs( z, val ).evalf()
outx = sympy.re( val ).evalf()
outy = sympy.im( val ).evalf()
del val
for disp in displays:
if (outx - disp.x)**2 + (outy - disp.y)**2 < 0.2:
out = (x+width/2,height-(y+height/2))
print out
screen.set_at( out, disp.c )
break
print "Vertical line %d complete" % ( x )
sympy.core.cache.clear_cache()
event = pygame.event.poll()
while event.type != pygame.NOEVENT:
if event.type == pygame.QUIT:
return False
event = pygame.event.poll()
pygame.display.flip()
def test_quaternion_complex_real_addition():
a = symbols("a", complex=True)
b = symbols("b", real=True)
# This symbol is not complex:
c = symbols("c", commutative=False)
q = Quaternion(x, y, z, w)
assert a + q == Quaternion(x + re(a), y + im(a), z, w)
assert 1 + q == Quaternion(1 + x, y, z, w)
assert I + q == Quaternion(x, 1 + y, z, w)
assert b + q == Quaternion(x + b, y, z, w)
assert c + q == Add(c, Quaternion(x, y, z, w), evaluate=False)
assert q * c == Mul(Quaternion(x, y, z, w), c, evaluate=False)
assert c * q == Mul(c, Quaternion(x, y, z, w), evaluate=False)
assert -q == Quaternion(-x, -y, -z, -w)
q1 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
q2 = Quaternion(1, 4, 7, 8)
assert q1 + (2 + 3*I) == Quaternion(5 + 7*I, 2 + 5*I, 0, 7 + 8*I)
assert q2 + (2 + 3*I) == Quaternion(3, 7, 7, 8)
assert q1 * (2 + 3*I) == \
Quaternion((2 + 3*I)*(3 + 4*I), (2 + 3*I)*(2 + 5*I), 0, (2 + 3*I)*(7 + 8*I))
assert q2 * (2 + 3*I) == Quaternion(-10, 11, 38, -5)
def test_Pow_is_real():
x = Symbol('x', real=True)
y = Symbol('y', real=True, positive=True)
assert (x**2).is_real is True
assert (x**3).is_real is True
assert (x**x).is_real is None
assert (y**x).is_real is True
assert (x**Rational(1, 3)).is_real is None
assert (y**Rational(1, 3)).is_real is True
assert sqrt(-1 - sqrt(2)).is_real is False
i = Symbol('i', imaginary=True)
assert (i**i).is_real is None
assert (I**i).is_real is None
assert ((-I)**i).is_real is None
assert (2**i).is_real is None # (2**(pi/log(2) * I)) is real, 2**I is not
assert (2**I).is_real is False
assert (2**-I).is_real is False
assert (i**2).is_real
assert (i**3).is_real is False
assert (i**x).is_real is None # could be (-I)**(2/3)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
k = Symbol('k', integer=True)
assert (i**e).is_real
assert (i**o).is_real is False
assert (i**k).is_real is None
x = Symbol("x", nonnegative=True)
y = Symbol("y", nonnegative=True)
assert im(x**y).expand(complex=True) is S.Zero
assert (x**y).is_real
def test_reduce_inequalities_multivariate():
assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == \
And(And(Or(Le(re(x), -1), Ge(re(x), 1)), Eq(im(x), 0)),
And(Or(Le(re(y), -1), Ge(re(y), 1)), Eq(im(y), 0)))
def reim(expr: Expr):
return re(expr), im(expr)
def test_coth():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert coth(nan) is nan
assert coth(zoo) is nan
assert coth(oo) == 1
assert coth(-oo) == -1
assert coth(0) is zoo
assert unchanged(coth, 1)
assert coth(-1) == -coth(1)
assert unchanged(coth, x)
assert coth(-x) == -coth(x)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == cot(pi) * I
assert unchanged(coth, 2**1024 * E)
assert coth(-2**1024 * E) == -coth(2**1024 * E)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == I * cot(pi)
assert coth(2 * pi * I) == -I * cot(2 * pi)
assert coth(-2 * pi * I) == I * cot(2 * pi)
assert coth(-3 * 10**73 * pi * I) == I * cot(3 * 10**73 * pi)
assert coth(7 * 10**103 * pi * I) == -I * cot(7 * 10**103 * pi)
assert coth(pi * I / 2) == 0
assert coth(-pi * I / 2) == 0
assert coth(pi * I * Rational(5, 2)) == 0
assert coth(pi * I * Rational(7, 2)) == 0
assert coth(pi * I / 3) == -I / sqrt(3)
assert coth(pi * I * Rational(-2, 3)) == -I / sqrt(3)
assert coth(pi * I / 4) == -I
assert coth(-pi * I / 4) == I
assert coth(pi * I * Rational(17, 4)) == -I
assert coth(pi * I * Rational(-3, 4)) == -I
assert coth(pi * I / 6) == -sqrt(3) * I
assert coth(-pi * I / 6) == sqrt(3) * I
assert coth(pi * I * Rational(7, 6)) == -sqrt(3) * I
assert coth(pi * I * Rational(-5, 6)) == -sqrt(3) * I
assert coth(pi * I / 105) == -cot(pi / 105) * I
assert coth(-pi * I / 105) == cot(pi / 105) * I
assert unchanged(coth, 2 + 3 * I)
assert coth(x * I) == -cot(x) * I
assert coth(k * pi * I) == -cot(k * pi) * I
assert coth(17 * k * pi * I) == -cot(17 * k * pi) * I
assert coth(k * pi * I) == -cot(k * pi) * I
assert coth(log(tan(2))) == coth(log(-tan(2)))
assert coth(1 + I * pi / 2) == tanh(1)
assert coth(x).as_real_imag(
deep=False) == (sinh(re(x)) * cosh(re(x)) /
(sin(im(x))**2 + sinh(re(x))**2), -sin(im(x)) *
cos(im(x)) / (sin(im(x))**2 + sinh(re(x))**2))
x = Symbol('x', extended_real=True)
assert coth(x).as_real_imag(deep=False) == (coth(x), 0)
assert expand_trig(coth(2 * x)) == (coth(x)**2 + 1) / (2 * coth(x))
assert expand_trig(coth(
3 * x)) == (coth(x)**3 + 3 * coth(x)) / (1 + 3 * coth(x)**2)
assert expand_trig(
coth(x + y)) == (1 + coth(x) * coth(y)) / (coth(x) + coth(y))
def test_fresnel():
assert fresnels(0) == 0
assert fresnels(oo) == S.Half
assert fresnels(-oo) == Rational(-1, 2)
assert fresnels(I * oo) == -I * S.Half
assert unchanged(fresnels, z)
assert fresnels(-z) == -fresnels(z)
assert fresnels(I * z) == -I * fresnels(z)
assert fresnels(-I * z) == I * fresnels(z)
assert conjugate(fresnels(z)) == fresnels(conjugate(z))
assert fresnels(z).diff(z) == sin(pi * z**2 / 2)
assert fresnels(z).rewrite(erf) == (S.One + I) / 4 * (erf(
(S.One + I) / 2 * sqrt(pi) * z) - I * erf(
(S.One - I) / 2 * sqrt(pi) * z))
assert fresnels(z).rewrite(hyper) == \
pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16)
assert fresnels(z).series(z, n=15) == \
pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15)
assert fresnels(w).is_extended_real is True
assert fresnels(w).is_finite is True
assert fresnels(z).is_extended_real is None
assert fresnels(z).is_finite is None
assert fresnels(z).as_real_imag() == (
fresnels(re(z) - I * im(z)) / 2 + fresnels(re(z) + I * im(z)) / 2,
-I * (-fresnels(re(z) - I * im(z)) + fresnels(re(z) + I * im(z))) / 2)
assert fresnels(z).as_real_imag(deep=False) == (
fresnels(re(z) - I * im(z)) / 2 + fresnels(re(z) + I * im(z)) / 2,
-I * (-fresnels(re(z) - I * im(z)) + fresnels(re(z) + I * im(z))) / 2)
assert fresnels(w).as_real_imag() == (fresnels(w), 0)
assert fresnels(w).as_real_imag(deep=True) == (fresnels(w), 0)
assert fresnels(2 + 3 * I).as_real_imag() == (
fresnels(2 + 3 * I) / 2 + fresnels(2 - 3 * I) / 2,
-I * (fresnels(2 + 3 * I) - fresnels(2 - 3 * I)) / 2)
assert expand_func(integrate(fresnels(z), z)) == \
z*fresnels(z) + cos(pi*z**2/2)/pi
assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(9, 4) * \
meijerg(((), (1,)), ((Rational(3, 4),),
(Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4))
assert fresnelc(0) == 0
assert fresnelc(oo) == S.Half
assert fresnelc(-oo) == Rational(-1, 2)
assert fresnelc(I * oo) == I * S.Half
assert unchanged(fresnelc, z)
assert fresnelc(-z) == -fresnelc(z)
assert fresnelc(I * z) == I * fresnelc(z)
assert fresnelc(-I * z) == -I * fresnelc(z)
assert conjugate(fresnelc(z)) == fresnelc(conjugate(z))
assert fresnelc(z).diff(z) == cos(pi * z**2 / 2)
assert fresnelc(z).rewrite(erf) == (S.One - I) / 4 * (erf(
(S.One + I) / 2 * sqrt(pi) * z) + I * erf(
(S.One - I) / 2 * sqrt(pi) * z))
assert fresnelc(z).rewrite(hyper) == \
z * hyper([Rational(1, 4)], [S.Half, Rational(5, 4)], -pi**2*z**4/16)
assert fresnelc(w).is_extended_real is True
assert fresnelc(z).as_real_imag() == \
(fresnelc(re(z) - I*im(z))/2 + fresnelc(re(z) + I*im(z))/2,
-I*(-fresnelc(re(z) - I*im(z)) + fresnelc(re(z) + I*im(z)))/2)
assert fresnelc(z).as_real_imag(deep=False) == \
(fresnelc(re(z) - I*im(z))/2 + fresnelc(re(z) + I*im(z))/2,
-I*(-fresnelc(re(z) - I*im(z)) + fresnelc(re(z) + I*im(z)))/2)
assert fresnelc(2 + 3 * I).as_real_imag() == (
fresnelc(2 - 3 * I) / 2 + fresnelc(2 + 3 * I) / 2,
-I * (fresnelc(2 + 3 * I) - fresnelc(2 - 3 * I)) / 2)
assert expand_func(integrate(fresnelc(z), z)) == \
z*fresnelc(z) - sin(pi*z**2/2)/pi
assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(3, 4) * \
meijerg(((), (1,)), ((Rational(1, 4),),
(Rational(3, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(1, 4)*(z**2)**Rational(1, 4))
from sympy.testing.randtest import verify_numerically
verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z)
verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z)
verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z)
verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z)
verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z)
verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z)
verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z)
verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
raises(ArgumentIndexError, lambda: fresnels(z).fdiff(2))
raises(ArgumentIndexError, lambda: fresnelc(z).fdiff(2))
assert fresnels(x).taylor_term(-1, x) is S.Zero
assert fresnelc(x).taylor_term(-1, x) is S.Zero
assert fresnelc(x).taylor_term(1, x) == -pi**2 * x**5 / 40
def test_expand_function():
assert expand(x + y) == x + y
assert expand(x + y, complex=True) == I * im(x) + I * im(y) + re(x) + re(y)
assert expand((x + y)**11, modulus=11) == x**11 + y**11
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3 * I) == I
assert sign(-3 * I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2 * I).doit() == sqrt(2) * (2 + 2 * I) / 4
assert sign(2 + 3 * I).simplify() == sign(2 + 3 * I)
assert sign(2 + 2 * I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_finite is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2 * x) == sign(x)
assert sign(2 * x) == sign(x)
assert sign(I * x) == I * sign(x)
assert sign(-2 * I * x) == -I * sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2 * p * x) == sign(x)
assert sign(n * x) == -sign(x)
assert sign(n * m * x) == sign(x)
x = Symbol('x', imaginary=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is False
assert sign(x).is_real is False
assert sign(x).is_zero is False
assert sign(x).diff(x) == 2 * DiracDelta(-I * x)
assert sign(x).doit() == x / Abs(x)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', real=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2 * DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(Abs(x)) == 0
assert Abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_real is True
assert sign(nz).is_zero is False
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', real=True)).is_nonnegative is None
assert sign(Symbol('x', real=True)).is_nonpositive is None
assert sign(Symbol('x', real=True, zero=False)).is_nonpositive is None
x, y = Symbol('x', real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
assert sign(x).rewrite(Heaviside) == 2 * Heaviside(x) - 1
assert sign(y).rewrite(Heaviside) == sign(y)
# evaluate what can be evaluated
assert sign(exp_polar(I * pi) * pi) is S.NegativeOne
eq = -sqrt(10 + 6 * sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3 * sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq).func is sign or sign(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2 * sqrt(3) + 1331 * sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert sign(d).func is sign or sign(d) == 0
def test_im():
x, y = symbols('x,y')
a, b = symbols('a,b', real=True)
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
assert im(nan) == nan
assert im(oo * I) == oo
assert im(-oo * I) == -oo
assert im(0) == 0
assert im(1) == 0
assert im(-1) == 0
assert im(E * I) == E
assert im(-E * I) == -E
assert im(x) == im(x)
assert im(x * I) == re(x)
assert im(r * I) == r
assert im(r) == 0
assert im(i * I) == 0
assert im(i) == -I * i
assert im(x + y) == im(x + y)
assert im(x + r) == im(x)
assert im(x + r * I) == im(x) + r
assert im(im(x) * I) == im(x)
assert im(2 + I) == 1
assert im(x + I) == im(x) + 1
assert im(x + y * I) == im(x) + re(y)
assert im(x + r * I) == im(x) + r
assert im(log(2 * I)) == pi / 2
assert im((2 + I)**2).expand(complex=True) == 4
assert im(conjugate(x)) == -im(x)
assert conjugate(im(x)) == im(x)
assert im(x).as_real_imag() == (im(x), 0)
assert im(i * r * x).diff(r) == im(i * x)
assert im(i * r * x).diff(i) == -I * re(r * x)
assert im(
sqrt(a +
b * I)) == (a**2 + b**2)**Rational(1, 4) * sin(atan2(b, a) / 2)
assert im(a * (2 + b * I)) == a * b
assert im((1 + sqrt(a + b*I))/2) == \
(a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2
assert im(x).rewrite(re) == x - re(x)
assert (x + im(y)).rewrite(im, re) == x + y - re(y)
def test_im():
x, y = symbols('x,y')
a, b = symbols('a,b', real=True)
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
assert im(nan) == nan
assert im(oo * I) == oo
assert im(-oo * I) == -oo
assert im(0) == 0
assert im(1) == 0
assert im(-1) == 0
assert im(E * I) == E
assert im(-E * I) == -E
assert unchanged(im, x)
assert im(x * I) == re(x)
assert im(r * I) == r
assert im(r) == 0
assert im(i * I) == 0
assert im(i) == -I * i
assert im(x + y) == im(x) + im(y)
assert im(x + r) == im(x)
assert im(x + r * I) == im(x) + r
assert im(im(x) * I) == im(x)
assert im(2 + I) == 1
assert im(x + I) == im(x) + 1
assert im(x + y * I) == im(x) + re(y)
assert im(x + r * I) == im(x) + r
assert im(log(2 * I)) == pi / 2
assert im((2 + I)**2).expand(complex=True) == 4
assert im(conjugate(x)) == -im(x)
assert conjugate(im(x)) == im(x)
assert im(x).as_real_imag() == (im(x), 0)
assert im(i * r * x).diff(r) == im(i * x)
assert im(i * r * x).diff(i) == -I * re(r * x)
assert im(
sqrt(a +
b * I)) == (a**2 + b**2)**Rational(1, 4) * sin(atan2(b, a) / 2)
assert im(a * (2 + b * I)) == a * b
assert im((1 + sqrt(a + b*I))/2) == \
(a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2
assert im(x).rewrite(re) == -S.ImaginaryUnit * (x - re(x))
assert (x + im(y)).rewrite(im, re) == x - S.ImaginaryUnit * (y - re(y))
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert im(S.ComplexInfinity) == S.NaN
n, m, l = symbols('n m l')
A = MatrixSymbol('A', n, m)
assert im(A) == (S(1) / (2 * I)) * (A - conjugate(A))
A = Matrix([[1 + 4 * I, 2], [0, -3 * I]])
assert im(A) == Matrix([[4, 0], [0, -3]])
A = ImmutableMatrix([[1 + 3 * I, 3 - 2 * I], [0, 2 * I]])
assert im(A) == ImmutableMatrix([[3, -2], [0, 2]])
X = ImmutableSparseMatrix([[i * I + i for i in range(5)]
for i in range(5)])
Y = SparseMatrix([[i for i in range(5)] for i in range(5)])
assert im(X).as_immutable() == Y
X = FunctionMatrix(3, 3, Lambda((n, m), n + m * I))
assert im(X) == Matrix([[0, 1, 2], [0, 1, 2], [0, 1, 2]])
def as_real_imag(self, deep=True, **hints):
from sympy import im, re
if hints.get('ignore') == self:
return None
else:
return (re(self), im(self))
def test_issue_12996():
# foo=True imitates the sort of arguments that Derivative can get
# from Integral when it passes doit to the expression
assert Derivative(im(x), x).doit(foo=True) == Derivative(im(x), x)
def test_Abs():
raises(TypeError, lambda: Abs(Interval(2, 3))) # issue 8717
x, y = symbols('x,y')
assert sign(sign(x)) == sign(x)
assert sign(x * y).func is sign
assert Abs(0) == 0
assert Abs(1) == 1
assert Abs(-1) == 1
assert Abs(I) == 1
assert Abs(-I) == 1
assert Abs(nan) == nan
assert Abs(zoo) == oo
assert Abs(I * pi) == pi
assert Abs(-I * pi) == pi
assert Abs(I * x) == Abs(x)
assert Abs(-I * x) == Abs(x)
assert Abs(-2 * x) == 2 * Abs(x)
assert Abs(-2.0 * x) == 2.0 * Abs(x)
assert Abs(2 * pi * x * y) == 2 * pi * Abs(x * y)
assert Abs(conjugate(x)) == Abs(x)
assert conjugate(Abs(x)) == Abs(x)
assert Abs(x).expand(complex=True) == sqrt(re(x)**2 + im(x)**2)
a = Symbol('a', positive=True)
assert Abs(2 * pi * x * a) == 2 * pi * a * Abs(x)
assert Abs(2 * pi * I * x * a) == 2 * pi * a * Abs(x)
x = Symbol('x', real=True)
n = Symbol('n', integer=True)
assert Abs((-1)**n) == 1
assert x**(2 * n) == Abs(x)**(2 * n)
assert Abs(x).diff(x) == sign(x)
assert abs(x) == Abs(x) # Python built-in
assert Abs(x)**3 == x**2 * Abs(x)
assert Abs(x)**4 == x**4
assert (Abs(x)**(3 * n)).args == (Abs(x), 3 * n
) # leave symbolic odd unchanged
assert (1 / Abs(x)).args == (Abs(x), -1)
assert 1 / Abs(x)**3 == 1 / (x**2 * Abs(x))
assert Abs(x)**-3 == Abs(x) / (x**4)
assert Abs(x**3) == x**2 * Abs(x)
assert Abs(I**I) == exp(-pi / 2)
assert Abs((4 + 5 * I)**(6 + 7 * I)) == 68921 * exp(-7 * atan(S(5) / 4))
y = Symbol('y', real=True)
assert Abs(I**y) == 1
y = Symbol('y')
assert Abs(I**y) == exp(-pi * im(y) / 2)
x = Symbol('x', imaginary=True)
assert Abs(x).diff(x) == -sign(x)
eq = -sqrt(10 + 6 * sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3 * sqrt(3))
# if there is a fast way to know when you can and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert abs(eq).func is Abs or abs(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2 * sqrt(3) + 1331 * sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert abs(d).func is Abs or abs(d) == 0
assert Abs(4 * exp(pi * I / 4)) == 4
assert Abs(3**(2 + I)) == 9
assert Abs((-3)**(1 - I)) == 3 * exp(pi)
assert Abs(oo) is oo
assert Abs(-oo) is oo
assert Abs(oo + I) is oo
assert Abs(oo + I * oo) is oo
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert Abs(x).fdiff() == sign(x)
raises(ArgumentIndexError, lambda: Abs(x).fdiff(2))
# doesn't have recursion error
arg = sqrt(acos(1 - I) * acos(1 + I))
assert abs(arg) == arg
# special handling to put Abs in denom
assert abs(1 / x) == 1 / Abs(x)
e = abs(2 / x**2)
assert e.is_Mul and e == 2 / Abs(x**2)
assert unchanged(Abs, y / x)
assert unchanged(Abs, x / (x + 1))
assert unchanged(Abs, x * y)
p = Symbol('p', positive=True)
assert abs(x / p) == abs(x) / p
# coverage
assert unchanged(Abs, Symbol('x', real=True)**y)
def test_cosh():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert cosh(nan) is nan
assert cosh(zoo) is nan
assert cosh(oo) is oo
assert cosh(-oo) is oo
assert cosh(0) == 1
assert unchanged(cosh, 1)
assert cosh(-1) == cosh(1)
assert unchanged(cosh, x)
assert cosh(-x) == cosh(x)
assert cosh(pi * I) == cos(pi)
assert cosh(-pi * I) == cos(pi)
assert unchanged(cosh, 2**1024 * E)
assert cosh(-2**1024 * E) == cosh(2**1024 * E)
assert cosh(pi * I / 2) == 0
assert cosh(-pi * I / 2) == 0
assert cosh((-3 * 10**73 + 1) * pi * I / 2) == 0
assert cosh((7 * 10**103 + 1) * pi * I / 2) == 0
assert cosh(pi * I) == -1
assert cosh(-pi * I) == -1
assert cosh(5 * pi * I) == -1
assert cosh(8 * pi * I) == 1
assert cosh(pi * I / 3) == S.Half
assert cosh(pi * I * Rational(-2, 3)) == Rational(-1, 2)
assert cosh(pi * I / 4) == S.Half * sqrt(2)
assert cosh(-pi * I / 4) == S.Half * sqrt(2)
assert cosh(pi * I * Rational(11, 4)) == Rational(-1, 2) * sqrt(2)
assert cosh(pi * I * Rational(-3, 4)) == Rational(-1, 2) * sqrt(2)
assert cosh(pi * I / 6) == S.Half * sqrt(3)
assert cosh(-pi * I / 6) == S.Half * sqrt(3)
assert cosh(pi * I * Rational(7, 6)) == Rational(-1, 2) * sqrt(3)
assert cosh(pi * I * Rational(-5, 6)) == Rational(-1, 2) * sqrt(3)
assert cosh(pi * I / 105) == cos(pi / 105)
assert cosh(-pi * I / 105) == cos(pi / 105)
assert unchanged(cosh, 2 + 3 * I)
assert cosh(x * I) == cos(x)
assert cosh(k * pi * I) == cos(k * pi)
assert cosh(17 * k * pi * I) == cos(17 * k * pi)
assert unchanged(cosh, k * pi)
assert cosh(x).as_real_imag(deep=False) == (cos(im(x)) * cosh(re(x)),
sin(im(x)) * sinh(re(x)))
x = Symbol('x', extended_real=True)
assert cosh(x).as_real_imag(deep=False) == (cosh(x), 0)
x = Symbol('x', real=True)
assert cosh(I * x).is_finite is True
assert cosh(I * x).is_real is True
assert cosh(I * 2 + 1).is_real is False
def test_re_im1652():
x = Symbol('x')
assert re(x) == re(conjugate(x))
assert im(x) == - im(conjugate(x))
assert im(x)*re(conjugate(x)) + im(conjugate(x)) * re(x) == 0
def test_polygamma():
from sympy import I
assert polygamma(n, nan) == nan
assert polygamma(0, oo) == oo
assert polygamma(0, -oo) == oo
assert polygamma(0, I*oo) == oo
assert polygamma(0, -I*oo) == oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) == zoo
assert polygamma(0, -9) == zoo
assert polygamma(0, -1) == zoo
assert polygamma(0, 0) == zoo
assert polygamma(0, 1) == -EulerGamma
assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
assert polygamma(1, 1) == pi**2/6
assert polygamma(1, 2) == pi**2/6 - 1
assert polygamma(1, 3) == pi**2/6 - Rational(5, 4)
assert polygamma(3, 1) == pi**4 / 15
assert polygamma(3, 5) == 6*(Rational(-22369, 20736) + pi**4/90)
assert polygamma(5, 1) == 8 * pi**6 / 63
def t(m, n):
x = S(m)/n
r = polygamma(0, x)
if r.has(polygamma):
return False
return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10
assert t(1, 2)
assert t(3, 2)
assert t(-1, 2)
assert t(1, 4)
assert t(-3, 4)
assert t(1, 3)
assert t(4, 3)
assert t(3, 4)
assert t(2, 3)
assert t(123, 5)
assert polygamma(0, x).rewrite(zeta) == polygamma(0, x)
assert polygamma(1, x).rewrite(zeta) == zeta(2, x)
assert polygamma(2, x).rewrite(zeta) == -2*zeta(3, x)
assert polygamma(I, 2).rewrite(zeta) == polygamma(I, 2)
n1 = Symbol('n1')
n2 = Symbol('n2', real=True)
n3 = Symbol('n3', integer=True)
n4 = Symbol('n4', positive=True)
n5 = Symbol('n5', positive=True, integer=True)
assert polygamma(n1, x).rewrite(zeta) == polygamma(n1, x)
assert polygamma(n2, x).rewrite(zeta) == polygamma(n2, x)
assert polygamma(n3, x).rewrite(zeta) == polygamma(n3, x)
assert polygamma(n4, x).rewrite(zeta) == polygamma(n4, x)
assert polygamma(n5, x).rewrite(zeta) == (-1)**(n5 + 1) * factorial(n5) * zeta(n5 + 1, x)
assert polygamma(3, 7*x).diff(x) == 7*polygamma(4, 7*x)
assert polygamma(0, x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
assert polygamma(2, x).rewrite(harmonic) == 2*harmonic(x - 1, 3) - 2*zeta(3)
ni = Symbol("n", integer=True)
assert polygamma(ni, x).rewrite(harmonic) == (-1)**(ni + 1)*(-harmonic(x - 1, ni + 1)
+ zeta(ni + 1))*factorial(ni)
# Polygamma of non-negative integer order is unbranched:
from sympy import exp_polar
k = Symbol('n', integer=True, nonnegative=True)
assert polygamma(k, exp_polar(2*I*pi)*x) == polygamma(k, x)
# but negative integers are branched!
k = Symbol('n', integer=True)
assert polygamma(k, exp_polar(2*I*pi)*x).args == (k, exp_polar(2*I*pi)*x)
# Polygamma of order -1 is loggamma:
assert polygamma(-1, x) == loggamma(x)
# But smaller orders are iterated integrals and don't have a special name
assert polygamma(-2, x).func is polygamma
# Test a bug
assert polygamma(0, -x).expand(func=True) == polygamma(0, -x)
assert polygamma(2, 2.5).is_positive == False
assert polygamma(2, -2.5).is_positive == False
assert polygamma(3, 2.5).is_positive == True
assert polygamma(3, -2.5).is_positive is None
assert polygamma(-2, -2.5).is_positive is None
assert polygamma(-3, -2.5).is_positive is None
assert polygamma(2, 2.5).is_negative == True
assert polygamma(3, 2.5).is_negative == False
assert polygamma(3, -2.5).is_negative == False
assert polygamma(2, -2.5).is_negative is None
assert polygamma(-2, -2.5).is_negative is None
assert polygamma(-3, -2.5).is_negative is None
assert polygamma(I, 2).is_positive is None
assert polygamma(I, 3).is_negative is None
# issue 17350
assert polygamma(pi, 3).evalf() == polygamma(pi, 3)
assert (I*polygamma(I, pi)).as_real_imag() == \
(-im(polygamma(I, pi)), re(polygamma(I, pi)))
assert (tanh(polygamma(I, 1))).rewrite(exp) == \
(exp(polygamma(I, 1)) - exp(-polygamma(I, 1)))/(exp(polygamma(I, 1)) + exp(-polygamma(I, 1)))
assert (I / polygamma(I, 4)).rewrite(exp) == \
I*sqrt(re(polygamma(I, 4))**2 + im(polygamma(I, 4))**2)\
/((re(polygamma(I, 4)) + I*im(polygamma(I, 4)))*Abs(polygamma(I, 4)))
assert unchanged(polygamma, 2.3, 1.0)
# issue 12569
assert unchanged(im, polygamma(0, I))
assert polygamma(Symbol('a', positive=True), Symbol('b', positive=True)).is_real is True
assert polygamma(0, I).is_real is None
def eval(cls, arg):
from sympy.calculus import AccumBounds
from sympy.sets.setexpr import SetExpr
from sympy.matrices.matrices import MatrixBase
from sympy import im, logcombine, re
if isinstance(arg, MatrixBase):
return arg.exp()
elif global_parameters.exp_is_pow:
return Pow(S.Exp1, arg)
elif arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.One
elif arg is S.One:
return S.Exp1
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.ComplexInfinity:
return S.NaN
elif isinstance(arg, log):
return arg.args[0]
elif isinstance(arg, AccumBounds):
return AccumBounds(exp(arg.min), exp(arg.max))
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
elif arg.is_Mul:
coeff = arg.as_coefficient(S.Pi*S.ImaginaryUnit)
if coeff:
if (2*coeff).is_integer:
if coeff.is_even:
return S.One
elif coeff.is_odd:
return S.NegativeOne
elif (coeff + S.Half).is_even:
return -S.ImaginaryUnit
elif (coeff + S.Half).is_odd:
return S.ImaginaryUnit
elif coeff.is_Rational:
ncoeff = coeff % 2 # restrict to [0, 2pi)
if ncoeff > 1: # restrict to (-pi, pi]
ncoeff -= 2
if ncoeff != coeff:
return cls(ncoeff*S.Pi*S.ImaginaryUnit)
# Warning: code in risch.py will be very sensitive to changes
# in this (see DifferentialExtension).
# look for a single log factor
coeff, terms = arg.as_coeff_Mul()
# but it can't be multiplied by oo
if coeff in [S.NegativeInfinity, S.Infinity]:
if terms.is_number:
if coeff is S.NegativeInfinity:
terms = -terms
if re(terms).is_zero and terms is not S.Zero:
return S.NaN
if re(terms).is_positive and im(terms) is not S.Zero:
return S.ComplexInfinity
if re(terms).is_negative:
return S.Zero
return None
coeffs, log_term = [coeff], None
for term in Mul.make_args(terms):
term_ = logcombine(term)
if isinstance(term_, log):
if log_term is None:
log_term = term_.args[0]
else:
return None
elif term.is_comparable:
coeffs.append(term)
else:
return None
return log_term**Mul(*coeffs) if log_term else None
elif arg.is_Add:
out = []
add = []
argchanged = False
for a in arg.args:
if a is S.One:
add.append(a)
continue
newa = cls(a)
if isinstance(newa, cls):
if newa.args[0] != a:
add.append(newa.args[0])
argchanged = True
else:
add.append(a)
else:
out.append(newa)
if out or argchanged:
return Mul(*out)*cls(Add(*add), evaluate=False)
if arg.is_zero:
return S.One
def inverse_laplace_ratfun(expr, s, t):
N, D, delay = Ratfun(expr, s).as_ratfun_delay()
# The delay should be zero
Q, M = sym.div(N, D, s)
result1 = sym.S.Zero
if Q:
Qpoly = sym.Poly(Q, s)
C = Qpoly.all_coeffs()
for n, c in enumerate(C):
result1 += c * sym.diff(sym.DiracDelta(t), t, len(C) - n - 1)
expr = M / D
for factor in expr.as_ordered_factors():
if factor == sym.oo:
return factor
sexpr = Ratfun(expr, s)
P = sexpr.poles()
result2 = sym.S.Zero
P2 = P.copy()
for p in P2:
# Number of occurrences of the pole.
N = P2[p]
if N == 0:
continue
f = s - p
if N == 1:
r = sexpr.residue(p, P)
pc = p.conjugate()
if pc != p and pc in P:
# Remove conjugate from poles and process pole with its
# conjugate. Unfortunately, for symbolic expressions
# we cannot tell if a quadratic has two real poles,
# a repeat real pole, or a complex conjugate pair of poles.
P2[pc] = 0
p_re = sym.re(p)
p_im = sym.im(p)
r_re = sym.re(r)
r_im = sym.im(r)
et = sym.exp(p_re * t)
result2 += 2 * r_re * et * sym.cos(p_im * t)
result2 -= 2 * r_im * et * sym.sin(p_im * t)
else:
result2 += r * sym.exp(p * t)
continue
# Handle repeated poles.
expr2 = expr * f ** N
for n in range(1, N + 1):
m = N - n
r = sym.limit(
sym.diff(expr2, s, m), s, p) / sym.factorial(m)
result2 += r * sym.exp(p * t) * t**(n - 1)
# result1 is a sum of Dirac deltas and its derivatives so is known
# to be causal.
return result1, result2
Uravnenie_all.append(expr2)
print(
"\nСистема уравнений или условия Куна-Таккера имеют следующий вид (при этом Nu1, Nu2 ,... >= 0 и все лямбды имеют произвольный знак)\n",
Uravnenie_all)
#Решение системы уравнений дает множество решений, далее мы их будем фильтровать
Solution = solve((Uravnenie_all), dict=True)
#избавляемся от решений с мнимыми числами
Non_imaginary = []
count_imag = 0
for i in range(len(Solution)):
count_imag = 0
for j in range(0, len(keys_lambda)):
if im(Solution[i][keys_lambda[j]]) != 0:
count_imag = count_imag + 1
if count_imag == 0:
Non_imaginary.append(Solution[i])
#Находим решения где все x и Nu >= 0
count_Nu = 0
Non_zero_Nu = []
for i in range(0, len(Non_imaginary)):
count_Nu = 0
for j in range(0, len(keys_Nu)):
if Non_imaginary[i][keys_Nu[j]] < 0:
count_Nu = count_Nu + 1
if count_Nu == 0:
Non_zero_Nu.append(Non_imaginary[i])
def test_sinh():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert sinh(nan) is nan
assert sinh(zoo) is nan
assert sinh(oo) is oo
assert sinh(-oo) is -oo
assert sinh(0) == 0
assert unchanged(sinh, 1)
assert sinh(-1) == -sinh(1)
assert unchanged(sinh, x)
assert sinh(-x) == -sinh(x)
assert unchanged(sinh, pi)
assert sinh(-pi) == -sinh(pi)
assert unchanged(sinh, 2**1024 * E)
assert sinh(-2**1024 * E) == -sinh(2**1024 * E)
assert sinh(pi * I) == 0
assert sinh(-pi * I) == 0
assert sinh(2 * pi * I) == 0
assert sinh(-2 * pi * I) == 0
assert sinh(-3 * 10**73 * pi * I) == 0
assert sinh(7 * 10**103 * pi * I) == 0
assert sinh(pi * I / 2) == I
assert sinh(-pi * I / 2) == -I
assert sinh(pi * I * Rational(5, 2)) == I
assert sinh(pi * I * Rational(7, 2)) == -I
assert sinh(pi * I / 3) == S.Half * sqrt(3) * I
assert sinh(pi * I * Rational(-2, 3)) == Rational(-1, 2) * sqrt(3) * I
assert sinh(pi * I / 4) == S.Half * sqrt(2) * I
assert sinh(-pi * I / 4) == Rational(-1, 2) * sqrt(2) * I
assert sinh(pi * I * Rational(17, 4)) == S.Half * sqrt(2) * I
assert sinh(pi * I * Rational(-3, 4)) == Rational(-1, 2) * sqrt(2) * I
assert sinh(pi * I / 6) == S.Half * I
assert sinh(-pi * I / 6) == Rational(-1, 2) * I
assert sinh(pi * I * Rational(7, 6)) == Rational(-1, 2) * I
assert sinh(pi * I * Rational(-5, 6)) == Rational(-1, 2) * I
assert sinh(pi * I / 105) == sin(pi / 105) * I
assert sinh(-pi * I / 105) == -sin(pi / 105) * I
assert unchanged(sinh, 2 + 3 * I)
assert sinh(x * I) == sin(x) * I
assert sinh(k * pi * I) == 0
assert sinh(17 * k * pi * I) == 0
assert sinh(k * pi * I / 2) == sin(k * pi / 2) * I
assert sinh(x).as_real_imag(deep=False) == (cos(im(x)) * sinh(re(x)),
sin(im(x)) * cosh(re(x)))
x = Symbol('x', extended_real=True)
assert sinh(x).as_real_imag(deep=False) == (sinh(x), 0)
x = Symbol('x', real=True)
assert sinh(I * x).is_finite is True
assert sinh(x).is_real is True
assert sinh(I).is_real is False
def test_erf():
assert erf(nan) is 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, x, evaluate=False)) == x # To cover code in erf
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_real is False
assert erf(0).is_real is True
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi)
assert erf(x * y).as_leading_term(y) == 2 * x * y / sqrt(pi)
assert (erf(x * y) / erf(y)).as_leading_term(y) == x
assert erf(1 / x).as_leading_term(x) == S.One
assert erf(z).rewrite('uppergamma') == sqrt(z**
2) * (1 - erfc(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], [Rational(-1, 2)], 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 limit(erf(x) / x, x, 0) == 2 / sqrt(pi)
assert limit(x**(-4) - sqrt(pi) * erf(x**2) / (2 * x**6), x, 0) == S(1) / 3
assert erf(x).as_real_imag() == \
(erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
-I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
assert erf(x).as_real_imag(deep=False) == \
(erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
-I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
assert erf(w).as_real_imag() == (erf(w), 0)
assert erf(w).as_real_imag(deep=False) == (erf(w), 0)
# issue 13575
assert erf(I).as_real_imag() == (0, -I * erf(I))
raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
assert erf(x).inverse() == erfinv
def test_tanh():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert tanh(nan) is nan
assert tanh(zoo) is nan
assert tanh(oo) == 1
assert tanh(-oo) == -1
assert tanh(0) == 0
assert unchanged(tanh, 1)
assert tanh(-1) == -tanh(1)
assert unchanged(tanh, x)
assert tanh(-x) == -tanh(x)
assert unchanged(tanh, pi)
assert tanh(-pi) == -tanh(pi)
assert unchanged(tanh, 2**1024 * E)
assert tanh(-2**1024 * E) == -tanh(2**1024 * E)
assert tanh(pi * I) == 0
assert tanh(-pi * I) == 0
assert tanh(2 * pi * I) == 0
assert tanh(-2 * pi * I) == 0
assert tanh(-3 * 10**73 * pi * I) == 0
assert tanh(7 * 10**103 * pi * I) == 0
assert tanh(pi * I / 2) is zoo
assert tanh(-pi * I / 2) is zoo
assert tanh(pi * I * Rational(5, 2)) is zoo
assert tanh(pi * I * Rational(7, 2)) is zoo
assert tanh(pi * I / 3) == sqrt(3) * I
assert tanh(pi * I * Rational(-2, 3)) == sqrt(3) * I
assert tanh(pi * I / 4) == I
assert tanh(-pi * I / 4) == -I
assert tanh(pi * I * Rational(17, 4)) == I
assert tanh(pi * I * Rational(-3, 4)) == I
assert tanh(pi * I / 6) == I / sqrt(3)
assert tanh(-pi * I / 6) == -I / sqrt(3)
assert tanh(pi * I * Rational(7, 6)) == I / sqrt(3)
assert tanh(pi * I * Rational(-5, 6)) == I / sqrt(3)
assert tanh(pi * I / 105) == tan(pi / 105) * I
assert tanh(-pi * I / 105) == -tan(pi / 105) * I
assert unchanged(tanh, 2 + 3 * I)
assert tanh(x * I) == tan(x) * I
assert tanh(k * pi * I) == 0
assert tanh(17 * k * pi * I) == 0
assert tanh(k * pi * I / 2) == tan(k * pi / 2) * I
assert tanh(x).as_real_imag(
deep=False) == (sinh(re(x)) * cosh(re(x)) /
(cos(im(x))**2 + sinh(re(x))**2), sin(im(x)) *
cos(im(x)) / (cos(im(x))**2 + sinh(re(x))**2))
x = Symbol('x', extended_real=True)
assert tanh(x).as_real_imag(deep=False) == (tanh(x), 0)
assert tanh(I * pi / 3 + 1).is_real is False
assert tanh(x).is_real is True
assert tanh(I * pi * x / 2).is_real is None
def test_issue_7450():
ans = integrate(exp(-(1 + I) * x), (x, 0, oo))
assert re(ans) == S.Half and im(ans) == -S.Half
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'
beta = Function('beta')
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(FallingFactorial(3,
k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\lvert{x}\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(polylog(x,
y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
assert latex(jacobi(n, a, b,
x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(
n, a, b,
x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a,
x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(
n, a,
x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n,
x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n,
x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a,
x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(
n, a,
x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a,
x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(
n, a,
x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
# Test latex printing of function names with "_"
assert latex(
polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
assert latex(polar_lift(0)**
3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
def getPhase(self):
self.H = self.solution[self.Vo]/self.solution[self.Vi]
self.H = simplify(self.H)
self.mod = simplify(sqrt(re(self.H)**2+im(self.H)**2))
self.phase = simplify(tan(im(self.H)/re(self.H))*180/3.14195)
return self.phase
def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import im, cancel, I, Order, logcombine
from itertools import product
if not logx:
logx = log(x)
if self.args[0] == x:
return logx
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k*x**l)
if r is not None:
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l*logx # XXX true regardless of assumptions?
return r
def coeff_exp(term, x):
coeff, exp = S.One, S.Zero
for factor in Mul.make_args(term):
if factor.has(x):
base, exp = factor.as_base_exp()
if base != x:
try:
return term.leadterm(x)
except ValueError:
return term, S.Zero
else:
coeff *= factor
return coeff, exp
# TODO new and probably slow
try:
a, b = arg.leadterm(x)
s = arg.nseries(x, n=n+b, logx=logx)
except (ValueError, NotImplementedError, PoleError):
s = arg.nseries(x, n=n, logx=logx)
while s.is_Order:
n += 1
s = arg.nseries(x, n=n, logx=logx)
a, b = s.removeO().leadterm(x)
p = cancel(s/(a*x**b) - 1).expand().powsimp()
if p.has(exp):
p = logcombine(p)
if isinstance(p, Order):
n = p.getn()
_, d = coeff_exp(p, x)
if not d.is_positive:
return log(a) + b*logx + Order(x**n, x)
def mul(d1, d2):
res = {}
for e1, e2 in product(d1, d2):
ex = e1 + e2
if ex < n:
res[ex] = res.get(ex, S.Zero) + d1[e1]*d2[e2]
return res
pterms = {}
for term in Add.make_args(p):
co1, e1 = coeff_exp(term, x)
pterms[e1] = pterms.get(e1, S.Zero) + co1.removeO()
k = S.One
terms = {}
pk = pterms
while k*d < n:
coeff = -(-1)**k/k
for ex in pk:
terms[ex] = terms.get(ex, S.Zero) + coeff*pk[ex]
pk = mul(pk, pterms)
k += S.One
res = log(a) + b*logx
for ex in terms:
res += terms[ex]*x**(ex)
if cdir != 0:
cdir = self.args[0].dir(x, cdir)
if a.is_real and a.is_negative and im(cdir) < 0:
res -= 2*I*S.Pi
return res + Order(x**n, x)
def ratfun(self, expr, s, t, **kwargs):
if kwargs.pop('pdb', False):
import pdb
pdb.set_trace()
sexpr = Ratfun(expr, s)
if kwargs.get('damped_sin', False):
if sexpr.degree == 2:
return self.do_damped_sin(sexpr, s, t)
# if False and sexpr.degree == 3 and Ratfun(expr * s).degree == 2:
# return self.do_damped_sin3(sexpr, s, t)
self.debug('Finding QRPO representation')
damping = kwargs.get('damping', None)
Q, R, P, O, delay, undef = sexpr.as_QRPO(damping)
if delay != 0:
# This will be caught and trigger expansion of the expression.
self.error('Unhandled delay %s' % delay)
cresult = Zero
if Q:
Qpoly = sym.Poly(Q, s)
C = Qpoly.all_coeffs()
for n, c in enumerate(C):
cresult += c * sym.diff(sym.DiracDelta(t), t, len(C) - n - 1)
if R == []:
return cresult, 0
uresult = 0
for m, (p, n, A) in enumerate(zip(P, O, R)):
# This is zero for the conjugate pole.
if R[m] == 0:
continue
# Search and remove conjugate pair.
has_conjpair = False
if p.is_complex and kwargs.get('pairs', True):
pc = p.conjugate()
Ac = A.conjugate()
for m2, p2 in enumerate(P[m + 1:]):
m2 += m + 1
if n == O[m2] and p2 == pc and R[m2] == Ac:
R[m2] = 0
has_conjpair = True
break
if has_conjpair:
# Combine conjugate pairs.
p = p.expand(complex=True)
A = A.expand(complex=True)
p_re = sym.re(p)
p_im = sym.im(p)
A_re = sym.re(A)
A_im = sym.im(A)
et = sym.exp(p_re * t)
result = 2 * A_re * et * sym.cos(p_im * t)
result -= 2 * A_im * et * sym.sin(p_im * t)
else:
result = A * sym.exp(p * t)
if n > 1:
result *= t**(n - 1) / sym.factorial(n - 1)
uresult += result
# cresult is a sum of Dirac deltas and its derivatives so is known
# to be causal.
return cresult, uresult
def test_issue_1985():
x = Symbol('x')
assert ((x + x*I)/(1 + I)).as_real_imag() == (re((x + I*x)/(1 + I)), im((x + I*x)/(1 + I)))
def test_expand_function():
assert expand(x + y) == x + y
assert expand(x + y, complex=True) == I * im(x) + I * im(y) + re(x) + re(y)
def test_reduce_inequalities_boolean():
assert reduce_inequalities([Eq(x**2, 0),
True]) == And(Eq(re(x), 0), Eq(im(x), 0))
assert reduce_inequalities([Eq(x**2, 0), False]) is False
def test_re():
x, y = symbols('x,y')
a, b = symbols('a,b', real=True)
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
assert re(nan) == nan
assert re(oo) == oo
assert re(-oo) == -oo
assert re(0) == 0
assert re(1) == 1
assert re(-1) == -1
assert re(E) == E
assert re(-E) == -E
assert unchanged(re, x)
assert re(x * I) == -im(x)
assert re(r * I) == 0
assert re(r) == r
assert re(i * I) == I * i
assert re(i) == 0
assert re(x + y) == re(x) + re(y)
assert re(x + r) == re(x) + r
assert re(re(x)) == re(x)
assert re(2 + I) == 2
assert re(x + I) == re(x)
assert re(x + y * I) == re(x) - im(y)
assert re(x + r * I) == re(x)
assert re(log(2 * I)) == log(2)
assert re((2 + I)**2).expand(complex=True) == 3
assert re(conjugate(x)) == re(x)
assert conjugate(re(x)) == re(x)
assert re(x).as_real_imag() == (re(x), 0)
assert re(i * r * x).diff(r) == re(i * x)
assert re(i * r * x).diff(i) == I * r * im(x)
assert re(
sqrt(a +
b * I)) == (a**2 + b**2)**Rational(1, 4) * cos(atan2(b, a) / 2)
assert re(a * (2 + b * I)) == 2 * a
assert re((1 + sqrt(a + b*I))/2) == \
(a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2)
assert re(x).rewrite(im) == x - S.ImaginaryUnit * im(x)
assert (x + re(y)).rewrite(re, im) == x + y - S.ImaginaryUnit * im(y)
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert re(S.ComplexInfinity) == S.NaN
n, m, l = symbols('n m l')
A = MatrixSymbol('A', n, m)
assert re(A) == (S(1) / 2) * (A + conjugate(A))
A = Matrix([[1 + 4 * I, 2], [0, -3 * I]])
assert re(A) == Matrix([[1, 2], [0, 0]])
A = ImmutableMatrix([[1 + 3 * I, 3 - 2 * I], [0, 2 * I]])
assert re(A) == ImmutableMatrix([[1, 3], [0, 0]])
X = SparseMatrix([[2 * j + i * I for i in range(5)] for j in range(5)])
assert re(X) - Matrix([[0, 0, 0, 0, 0], [2, 2, 2, 2, 2], [4, 4, 4, 4, 4],
[6, 6, 6, 6, 6], [8, 8, 8, 8, 8]
]) == Matrix.zeros(5)
assert im(X) - Matrix([[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [0, 1, 2, 3, 4],
[0, 1, 2, 3, 4], [0, 1, 2, 3, 4]
]) == Matrix.zeros(5)
X = FunctionMatrix(3, 3, Lambda((n, m), n + m * I))
assert re(X) == Matrix([[0, 0, 0], [1, 1, 1], [2, 2, 2]])
def PlotDensityMatrix(self, imaginary=False, decimal=False):
size_p = self.getNumberOfQubits() # número de qubits
mRho = [[0] * 2**size_p for i in range(2**size_p)]
rho = self.DensityMatrix().get()
for id1 in range(2**size_p):
for id2 in range(2**size_p):
if self.getCuda():
value = rho[id1][id2]
if (not imaginary):
value = value.item().real
else:
value = value.item().imag
else:
value = rho[id1, id2]
if (not imaginary):
value = sp.re(value)
else:
value = sp.im(value)
mRho[id1][id2] = value
result = np.array(mRho, dtype=np.float)
fig = plt.figure(figsize=(5, 5), dpi=150)
ax1 = fig.add_subplot(111, projection='3d')
size = 2**size_p
if (not decimal):
labels = [
"|" + "{0:b}".format(i).zfill(size_p) + ">"
for i in range(size)
]
else:
labels = [i for i in range(size)]
xlabels = np.array(labels)
xpos = np.arange(xlabels.shape[0])
ylabels = np.array(labels)
ypos = np.arange(ylabels.shape[0])
xpos_mesh, ypos_mesh = np.meshgrid(xpos, ypos, copy=False)
zpos = result
zpos = zpos.ravel()
dx = 0.5
dy = 0.5
dz = zpos
ax1.w_xaxis.set_ticks(xpos + dx / 2.)
ax1.w_xaxis.set_ticklabels(xlabels)
ax1.w_yaxis.set_ticks(ypos + dy / 2.)
ax1.w_yaxis.set_ticklabels(ylabels)
values = np.linspace(0.2, 1., xpos_mesh.ravel().shape[0])
colors = cm.rainbow(values)
ax1.bar3d(xpos_mesh.ravel(),
ypos_mesh.ravel(),
dz * 0,
dx,
dy,
dz,
color=colors)
plt.show()
