def test_heaviside():
assert Heaviside(-5) == 0
assert Heaviside(1) == 1
assert Heaviside(0) == S.Half
assert Heaviside(0, x) == x
assert unchanged(Heaviside,x, nan)
assert Heaviside(0, nan) == nan
h0 = Heaviside(x, 0)
h12 = Heaviside(x, S.Half)
h1 = Heaviside(x, 1)
assert h0.args == h0.pargs == (x, 0)
assert h1.args == h1.pargs == (x, 1)
assert h12.args == (x, S.Half)
assert h12.pargs == (x,) # default 1/2 suppressed
assert adjoint(Heaviside(x)) == Heaviside(x)
assert adjoint(Heaviside(x - y)) == Heaviside(x - y)
assert conjugate(Heaviside(x)) == Heaviside(x)
assert conjugate(Heaviside(x - y)) == Heaviside(x - y)
assert transpose(Heaviside(x)) == Heaviside(x)
assert transpose(Heaviside(x - y)) == Heaviside(x - y)
assert Heaviside(x).diff(x) == DiracDelta(x)
assert Heaviside(x + I).is_Function is True
assert Heaviside(I*x).is_Function is True
raises(ArgumentIndexError, lambda: Heaviside(x).fdiff(2))
raises(ValueError, lambda: Heaviside(I))
raises(ValueError, lambda: Heaviside(2 + 3*I))
def test_hermite():
assert hermite(0, x) == 1
assert hermite(1, x) == 2 * x
assert hermite(2, x) == 4 * x**2 - 2
assert hermite(3, x) == 8 * x**3 - 12 * x
assert hermite(4, x) == 16 * x**4 - 48 * x**2 + 12
assert hermite(6, x) == 64 * x**6 - 480 * x**4 + 720 * x**2 - 120
n = Symbol("n")
assert unchanged(hermite, n, x)
assert hermite(n, -x) == (-1)**n * hermite(n, x)
assert unchanged(hermite, -n, x)
assert hermite(n, 0) == 2**n * sqrt(pi) / gamma(S.Half - n / 2)
assert hermite(n, oo) is oo
assert conjugate(hermite(n, x)) == hermite(n, conjugate(x))
_k = Dummy('k')
assert hermite(n, x).rewrite("polynomial").dummy_eq(
factorial(n) * Sum((-1)**_k * (2 * x)**(-2 * _k + n) /
(factorial(_k) * factorial(-2 * _k + n)),
(_k, 0, floor(n / 2))))
assert diff(hermite(n, x), x) == 2 * n * hermite(n - 1, x)
assert diff(hermite(n, x), n) == Derivative(hermite(n, x), n)
raises(ArgumentIndexError, lambda: hermite(n, x).fdiff(3))
def test_issue_11518():
x = Symbol("x", real=True)
y = Symbol("y", real=True)
r = sqrt(x**2 + y**2)
assert conjugate(r) == r
s = abs(x + I * y)
assert conjugate(s) == r
def test_multigamma():
from sympy.concrete.products import Product
p = Symbol('p')
_k = Dummy('_k')
assert multigamma(x, p).dummy_eq(pi**(p*(p - 1)/4)*\
Product(gamma(x + (1 - _k)/2), (_k, 1, p)))
assert conjugate(multigamma(x, p)).dummy_eq(pi**((conjugate(p) - 1)*\
conjugate(p)/4)*Product(gamma(conjugate(x) + (1-conjugate(_k))/2), (_k, 1, p)))
assert conjugate(multigamma(x, 1)) == gamma(conjugate(x))
p = Symbol('p', positive=True)
assert conjugate(multigamma(x, p)).dummy_eq(pi**((p - 1)*p/4)*\
Product(gamma(conjugate(x) + (1-conjugate(_k))/2), (_k, 1, p)))
assert multigamma(nan, 1) is nan
assert multigamma(oo, 1).doit() is oo
assert multigamma(1, 1) == 1
assert multigamma(2, 1) == 1
assert multigamma(3, 1) == 2
assert multigamma(102, 1) == factorial(101)
assert multigamma(S.Half, 1) == sqrt(pi)
assert multigamma(1, 2) == pi
assert multigamma(2, 2) == pi/2
assert multigamma(1, 3) is zoo
assert multigamma(2, 3) == pi**2/2
assert multigamma(3, 3) == 3*pi**2/2
assert multigamma(x, 1).diff(x) == gamma(x)*polygamma(0, x)
assert multigamma(x, 2).diff(x) == sqrt(pi)*gamma(x)*gamma(x - S.Half)*\
polygamma(0, x) + sqrt(pi)*gamma(x)*gamma(x - S.Half)*polygamma(0, x - S.Half)
assert multigamma(x - 1, 1).expand(func=True) == gamma(x)/(x - 1)
assert multigamma(x + 2, 1).expand(func=True, mul=False) == x*(x + 1)*\
gamma(x)
assert multigamma(x - 1, 2).expand(func=True) == sqrt(pi)*gamma(x)*\
gamma(x + S.Half)/(x**3 - 3*x**2 + x*Rational(11, 4) - Rational(3, 4))
assert multigamma(x - 1, 3).expand(func=True) == pi**Rational(3, 2)*gamma(x)**2*\
gamma(x + S.Half)/(x**5 - 6*x**4 + 55*x**3/4 - 15*x**2 + x*Rational(31, 4) - Rational(3, 2))
assert multigamma(n, 1).rewrite(factorial) == factorial(n - 1)
assert multigamma(n, 2).rewrite(factorial) == sqrt(pi)*\
factorial(n - Rational(3, 2))*factorial(n - 1)
assert multigamma(n, 3).rewrite(factorial) == pi**Rational(3, 2)*\
factorial(n - 2)*factorial(n - Rational(3, 2))*factorial(n - 1)
assert multigamma(Rational(-1, 2), 3, evaluate=False).is_real == False
assert multigamma(S.Half, 3, evaluate=False).is_real == False
assert multigamma(0, 1, evaluate=False).is_real == False
assert multigamma(1, 3, evaluate=False).is_real == False
assert multigamma(-1.0, 3, evaluate=False).is_real == False
assert multigamma(0.7, 3, evaluate=False).is_real == True
assert multigamma(3, 3, evaluate=False).is_real == True
def test_airy_base():
z = Symbol('z')
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert conjugate(airyai(z)) == airyai(conjugate(z))
assert airyai(x).is_extended_real
assert airyai(x+I*y).as_real_imag() == (
airyai(x - I*y)/2 + airyai(x + I*y)/2,
I*(airyai(x - I*y) - airyai(x + I*y))/2)
def test_erfi():
assert erfi(nan) is nan
assert erfi(oo) is S.Infinity
assert erfi(-oo) is S.NegativeInfinity
assert erfi(0) is S.Zero
assert erfi(I*oo) == I
assert erfi(-I*oo) == -I
assert erfi(-x) == -erfi(x)
assert erfi(I*erfinv(x)) == I*x
assert erfi(I*erfcinv(x)) == I*(1 - x)
assert erfi(I*erf2inv(0, x)) == I*x
assert erfi(I*erf2inv(0, x, evaluate=False)) == I*x # To cover code in erfi
assert erfi(I).is_real is False
assert erfi(0, evaluate=False).is_real
assert erfi(0, evaluate=False).is_zero
assert conjugate(erfi(z)) == erfi(conjugate(z))
assert erfi(x).as_leading_term(x) == 2*x/sqrt(pi)
assert erfi(x*y).as_leading_term(y) == 2*x*y/sqrt(pi)
assert (erfi(x*y)/erfi(y)).as_leading_term(y) == x
assert erfi(1/x).as_leading_term(x) == erfi(1/x)
assert erfi(z).rewrite('erf') == -I*erf(I*z)
assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I
assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi)
assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], -z**2)/sqrt(pi)
assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half,
-z**2)/sqrt(S.Pi) - S.One))
assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi)
assert erfi(z).rewrite('tractable') == -I*(-_erfs(I*z)*exp(z**2) + 1)
assert expand_func(erfi(I*z)) == I*erf(z)
assert erfi(x).as_real_imag() == \
(erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2,
-I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2)
assert erfi(x).as_real_imag(deep=False) == \
(erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2,
-I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2)
assert erfi(w).as_real_imag() == (erfi(w), 0)
assert erfi(w).as_real_imag(deep=False) == (erfi(w), 0)
raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def test_legendre():
assert legendre(0, x) == 1
assert legendre(1, x) == x
assert legendre(2, x) == ((3 * x**2 - 1) / 2).expand()
assert legendre(3, x) == ((5 * x**3 - 3 * x) / 2).expand()
assert legendre(4, x) == ((35 * x**4 - 30 * x**2 + 3) / 8).expand()
assert legendre(5, x) == ((63 * x**5 - 70 * x**3 + 15 * x) / 8).expand()
assert legendre(6, x) == ((231 * x**6 - 315 * x**4 + 105 * x**2 - 5) /
16).expand()
assert legendre(10, -1) == 1
assert legendre(11, -1) == -1
assert legendre(10, 1) == 1
assert legendre(11, 1) == 1
assert legendre(10, 0) != 0
assert legendre(11, 0) == 0
assert legendre(-1, x) == 1
k = Symbol('k')
assert legendre(5 - k, x).subs(k, 2) == ((5 * x**3 - 3 * x) / 2).expand()
assert roots(legendre(4, x), x) == {
sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
}
n = Symbol("n")
X = legendre(n, x)
assert isinstance(X, legendre)
assert unchanged(legendre, n, x)
assert legendre(n,
0) == sqrt(pi) / (gamma(S.Half - n / 2) * gamma(n / 2 + 1))
assert legendre(n, 1) == 1
assert legendre(n, oo) is oo
assert legendre(-n, x) == legendre(n - 1, x)
assert legendre(n, -x) == (-1)**n * legendre(n, x)
assert unchanged(legendre, -n + k, x)
assert conjugate(legendre(n, x)) == legendre(n, conjugate(x))
assert diff(legendre(n, x), x) == \
n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
assert diff(legendre(n, x), n) == Derivative(legendre(n, x), n)
_k = Dummy('k')
assert legendre(n, x).rewrite("polynomial").dummy_eq(
Sum((-1)**_k * (S.Half - x / 2)**_k * (x / 2 + S.Half)**(-_k + n) *
binomial(n, _k)**2, (_k, 0, n)))
raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(1))
raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(3))
def test_gegenbauer():
n = Symbol("n")
a = Symbol("a")
assert gegenbauer(0, a, x) == 1
assert gegenbauer(1, a, x) == 2 * a * x
assert gegenbauer(2, a, x) == -a + x**2 * (2 * a**2 + 2 * a)
assert gegenbauer(3, a, x) == \
x**3*(4*a**3/3 + 4*a**2 + a*Rational(8, 3)) + x*(-2*a**2 - 2*a)
assert gegenbauer(-1, a, x) == 0
assert gegenbauer(n, S.Half, x) == legendre(n, x)
assert gegenbauer(n, 1, x) == chebyshevu(n, x)
assert gegenbauer(n, -1, x) == 0
X = gegenbauer(n, a, x)
assert isinstance(X, gegenbauer)
assert gegenbauer(n, a, -x) == (-1)**n * gegenbauer(n, a, x)
assert gegenbauer(n, a, 0) == 2**n*sqrt(pi) * \
gamma(a + n/2)/(gamma(a)*gamma(-n/2 + S.Half)*gamma(n + 1))
assert gegenbauer(n, a,
1) == gamma(2 * a + n) / (gamma(2 * a) * gamma(n + 1))
assert gegenbauer(n, Rational(3, 4), -1) is zoo
assert gegenbauer(n, Rational(1, 4),
-1) == (sqrt(2) * cos(pi * (n + S.One / 4)) *
gamma(n + S.Half) / (sqrt(pi) * gamma(n + 1)))
m = Symbol("m", positive=True)
assert gegenbauer(m, a, oo) == oo * RisingFactorial(a, m)
assert unchanged(gegenbauer, n, a, oo)
assert conjugate(gegenbauer(n,
a, x)) == gegenbauer(n, conjugate(a),
conjugate(x))
_k = Dummy('k')
assert diff(gegenbauer(n, a, x), n) == Derivative(gegenbauer(n, a, x), n)
assert diff(gegenbauer(n, a, x), a).dummy_eq(
Sum((2 * (-1)**(-_k + n) + 2) * (_k + a) * gegenbauer(_k, a, x) /
((-_k + n) * (_k + 2 * a + n)) +
((2 * _k + 2) / ((_k + 2 * a) * (2 * _k + 2 * a + 1)) + 2 /
(_k + 2 * a + n)) * gegenbauer(n, a, x), (_k, 0, n - 1)))
assert diff(gegenbauer(n, a, x), x) == 2 * a * gegenbauer(n - 1, a + 1, x)
assert gegenbauer(n, a, x).rewrite('polynomial').dummy_eq(
Sum((-1)**_k * (2 * x)**(-2 * _k + n) * RisingFactorial(a, -_k + n) /
(factorial(_k) * factorial(-2 * _k + n)), (_k, 0, floor(n / 2))))
raises(ArgumentIndexError, lambda: gegenbauer(n, a, x).fdiff(4))
def test_transpose():
assert transpose(A).is_commutative is False
assert transpose(A*A) == transpose(A)**2
assert transpose(A*B) == transpose(B)*transpose(A)
assert transpose(A*B**2) == transpose(B)**2*transpose(A)
assert transpose(A*B - B*A) == \
transpose(B)*transpose(A) - transpose(A)*transpose(B)
assert transpose(A + I*B) == transpose(A) + I*transpose(B)
assert transpose(X) == conjugate(X)
assert transpose(-I*X) == -I*conjugate(X)
assert transpose(Y) == -conjugate(Y)
assert transpose(-I*Y) == I*conjugate(Y)
def test_assoc_legendre():
Plm = assoc_legendre
Q = sqrt(1 - x**2)
assert Plm(0, 0, x) == 1
assert Plm(1, 0, x) == x
assert Plm(1, 1, x) == -Q
assert Plm(2, 0, x) == (3 * x**2 - 1) / 2
assert Plm(2, 1, x) == -3 * x * Q
assert Plm(2, 2, x) == 3 * Q**2
assert Plm(3, 0, x) == (5 * x**3 - 3 * x) / 2
assert Plm(3, 1,
x).expand() == ((3 * (1 - 5 * x**2) / 2).expand() * Q).expand()
assert Plm(3, 2, x) == 15 * x * Q**2
assert Plm(3, 3, x) == -15 * Q**3
# negative m
assert Plm(1, -1, x) == -Plm(1, 1, x) / 2
assert Plm(2, -2, x) == Plm(2, 2, x) / 24
assert Plm(2, -1, x) == -Plm(2, 1, x) / 6
assert Plm(3, -3, x) == -Plm(3, 3, x) / 720
assert Plm(3, -2, x) == Plm(3, 2, x) / 120
assert Plm(3, -1, x) == -Plm(3, 1, x) / 12
n = Symbol("n")
m = Symbol("m")
X = Plm(n, m, x)
assert isinstance(X, assoc_legendre)
assert Plm(n, 0, x) == legendre(n, x)
assert Plm(n, m, 0) == 2**m * sqrt(pi) / (gamma(-m / 2 - n / 2 + S.Half) *
gamma(-m / 2 + n / 2 + 1))
assert diff(Plm(m, n, x),
x) == (m * x * assoc_legendre(m, n, x) -
(m + n) * assoc_legendre(m - 1, n, x)) / (x**2 - 1)
_k = Dummy('k')
assert Plm(m, n, x).rewrite("polynomial").dummy_eq(
(1 - x**2)**(n / 2) * Sum(
(-1)**_k * 2**(-m) * x**(-2 * _k + m - n) *
factorial(-2 * _k + 2 * m) /
(factorial(_k) * factorial(-_k + m) * factorial(-2 * _k + m - n)),
(_k, 0, floor(m / 2 - n / 2))))
assert conjugate(assoc_legendre(n, m, x)) == \
assoc_legendre(n, conjugate(m), conjugate(x))
raises(ValueError, lambda: Plm(0, 1, x))
raises(ValueError, lambda: Plm(-1, 1, x))
raises(ArgumentIndexError, lambda: Plm(n, m, x).fdiff(1))
raises(ArgumentIndexError, lambda: Plm(n, m, x).fdiff(2))
raises(ArgumentIndexError, lambda: Plm(n, m, x).fdiff(4))
def test_erfc():
assert erfc(nan) is nan
assert erfc(oo) is S.Zero
assert erfc(-oo) == 2
assert erfc(0) == 1
assert erfc(I*oo) == -oo*I
assert erfc(-I*oo) == oo*I
assert erfc(-x) == S(2) - erfc(x)
assert erfc(erfcinv(x)) == x
assert erfc(I).is_real is False
assert erfc(0, evaluate=False).is_real
assert erfc(0, evaluate=False).is_zero is False
assert erfc(erfinv(x)) == 1 - x
assert conjugate(erfc(z)) == erfc(conjugate(z))
assert erfc(x).as_leading_term(x) is S.One
assert erfc(1/x).as_leading_term(x) == S.Zero
assert erfc(z).rewrite('erf') == 1 - erf(z)
assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z)
assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi)
assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi)
assert erfc(z).rewrite('tractable') == _erfs(z)*exp(-z**2)
assert expand_func(erf(x) + erfc(x)) is S.One
assert erfc(x).as_real_imag() == \
(erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2,
-I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2)
assert erfc(x).as_real_imag(deep=False) == \
(erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2,
-I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2)
assert erfc(w).as_real_imag() == (erfc(w), 0)
assert erfc(w).as_real_imag(deep=False) == (erfc(w), 0)
raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
assert erfc(x).inverse() == erfcinv
def norm(self):
"""
Return the normalization of the specified functional form.
This function integrates over the coordinates of the Wavefunction, with
the bounds specified.
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
1
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
sqrt(2)*sqrt(L)/2
"""
exp = self.expr * conjugate(self.expr)
var = self.variables
limits = self.limits
for v in var:
curr_limits = limits[v]
exp = integrate(exp, (v, curr_limits[0], curr_limits[1]))
return sqrt(exp)
def inverse(self):
"""Returns the inverse of the quaternion."""
q = self
if not q.norm():
raise ValueError(
"Cannot compute inverse for a quaternion with zero norm")
return conjugate(q) * (1 / q.norm()**2)
def test_scalars():
x = symbols('x', complex=True)
assert Dagger(x) == conjugate(x)
assert Dagger(I * x) == -I * conjugate(x)
i = symbols('i', real=True)
assert Dagger(i) == i
p = symbols('p')
assert isinstance(Dagger(p), adjoint)
i = Integer(3)
assert Dagger(i) == i
A = symbols('A', commutative=False)
assert Dagger(A).is_commutative is False
def test_uppergamma():
from sympy.functions.special.error_functions import expint
from sympy.functions.special.hyper import meijerg
assert uppergamma(4, 0) == 6
assert uppergamma(x, y).diff(y) == -y**(x - 1)*exp(-y)
assert td(uppergamma(randcplx(), y), y)
assert uppergamma(x, y).diff(x) == \
uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
assert td(uppergamma(x, randcplx()), x)
p = Symbol('p', positive=True)
assert uppergamma(0, p) == -Ei(-p)
assert uppergamma(p, 0) == gamma(p)
assert uppergamma(S.Half, x) == sqrt(pi)*erfc(sqrt(x))
assert not uppergamma(S.Half - 3, x).has(uppergamma)
assert not uppergamma(S.Half + 3, x).has(uppergamma)
assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
assert tn(uppergamma(S.Half + 3, x, evaluate=False),
uppergamma(S.Half + 3, x), x)
assert tn(uppergamma(S.Half - 3, x, evaluate=False),
uppergamma(S.Half - 3, x), x)
assert unchanged(uppergamma, x, -oo)
assert unchanged(uppergamma, x, 0)
assert tn_branch(-3, uppergamma)
assert tn_branch(-4, uppergamma)
assert tn_branch(Rational(1, 3), uppergamma)
assert tn_branch(pi, uppergamma)
assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x)
assert uppergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + \
gamma(y)*(1 - exp(4*pi*I*y))
assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I
assert uppergamma(-2, x) == expint(3, x)/x**2
assert conjugate(uppergamma(x, y)) == uppergamma(conjugate(x), conjugate(y))
assert unchanged(conjugate, uppergamma(x, -oo))
assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)
assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)
assert uppergamma(70, 6) == 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320*exp(-6)
assert (uppergamma(S(77) / 2, 6) - uppergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
assert (uppergamma(-S(77) / 2, 6) - uppergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
def test_adjoint():
assert adjoint(A).is_commutative is False
assert adjoint(A*A) == adjoint(A)**2
assert adjoint(A*B) == adjoint(B)*adjoint(A)
assert adjoint(A*B**2) == adjoint(B)**2*adjoint(A)
assert adjoint(A*B - B*A) == adjoint(B)*adjoint(A) - adjoint(A)*adjoint(B)
assert adjoint(A + I*B) == adjoint(A) - I*adjoint(B)
assert adjoint(X) == X
assert adjoint(-I*X) == I*X
assert adjoint(Y) == -Y
assert adjoint(-I*Y) == -I*Y
assert adjoint(X) == conjugate(transpose(X))
assert adjoint(Y) == conjugate(transpose(Y))
assert adjoint(X) == transpose(conjugate(X))
assert adjoint(Y) == transpose(conjugate(Y))
def test_arg():
assert arg(0) is nan
assert arg(1) == 0
assert arg(-1) == pi
assert arg(I) == pi / 2
assert arg(-I) == -pi / 2
assert arg(1 + I) == pi / 4
assert arg(-1 + I) == pi * Rational(3, 4)
assert arg(1 - I) == -pi / 4
assert arg(exp_polar(4 * pi * I)) == 4 * pi
assert arg(exp_polar(-7 * pi * I)) == -7 * pi
assert arg(exp_polar(5 - 3 * pi * I / 4)) == pi * Rational(-3, 4)
f = Function('f')
assert not arg(f(0) + I * f(1)).atoms(re)
# check nesting
x = Symbol('x')
assert arg(arg(arg(x))) is not S.NaN
assert arg(arg(arg(arg(x)))) is S.NaN
r = Symbol('r', extended_real=True)
assert arg(arg(r)) is not S.NaN
assert arg(arg(arg(r))) is S.NaN
p = Function('p', extended_positive=True)
assert arg(p(x)) == 0
assert arg((3 + I) * p(x)) == arg(3 + I)
p = Symbol('p', positive=True)
assert arg(p) == 0
assert arg(p * I) == pi / 2
n = Symbol('n', negative=True)
assert arg(n) == pi
assert arg(n * I) == -pi / 2
x = Symbol('x')
assert conjugate(arg(x)) == arg(x)
e = p + I * p**2
assert arg(e) == arg(1 + p * I)
# make sure sign doesn't swap
e = -2 * p + 4 * I * p**2
assert arg(e) == arg(-1 + 2 * p * I)
# make sure sign isn't lost
x = symbols('x', real=True) # could be zero
e = x + I * x
assert arg(e) == arg(x * (1 + I))
assert arg(e / p) == arg(x * (1 + I))
e = p * cos(p) + I * log(p) * exp(p)
assert arg(e).args[0] == e
# keep it simple -- let the user do more advanced cancellation
e = (p + 1) + I * (p**2 - 1)
assert arg(e).args[0] == e
f = Function('f')
e = 2 * x * (f(0) - 1) - 2 * x * f(0)
assert arg(e) == arg(-2 * x)
assert arg(f(0)).func == arg and arg(f(0)).args == (f(0), )
def test_erf2():
assert erf2(0, 0) is S.Zero
assert erf2(x, x) is S.Zero
assert erf2(nan, 0) is nan
assert erf2(-oo, y) == erf(y) + 1
assert erf2( oo, y) == erf(y) - 1
assert erf2( x, oo) == 1 - erf(x)
assert erf2( x,-oo) == -1 - erf(x)
assert erf2(x, erf2inv(x, y)) == y
assert erf2(-x, -y) == -erf2(x,y)
assert erf2(-x, y) == erf(y) + erf(x)
assert erf2( x, -y) == -erf(y) - erf(x)
assert erf2(x, y).rewrite('fresnels') == erf(y).rewrite(fresnels)-erf(x).rewrite(fresnels)
assert erf2(x, y).rewrite('fresnelc') == erf(y).rewrite(fresnelc)-erf(x).rewrite(fresnelc)
assert erf2(x, y).rewrite('hyper') == erf(y).rewrite(hyper)-erf(x).rewrite(hyper)
assert erf2(x, y).rewrite('meijerg') == erf(y).rewrite(meijerg)-erf(x).rewrite(meijerg)
assert erf2(x, y).rewrite('uppergamma') == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
assert erf2(x, y).rewrite('expint') == erf(y).rewrite(expint)-erf(x).rewrite(expint)
assert erf2(I, 0).is_real is False
assert erf2(0, 0, evaluate=False).is_real
assert erf2(0, 0, evaluate=False).is_zero
assert erf2(x, x, evaluate=False).is_zero
assert erf2(x, y).is_zero is None
assert expand_func(erf(x) + erf2(x, y)) == erf(y)
assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))
assert erf2(x, y).rewrite('erf') == erf(y) - erf(x)
assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
assert erf2(x, y).rewrite('erfi') == I*(erfi(I*x) - erfi(I*y))
assert erf2(x, y).diff(x) == erf2(x, y).fdiff(1)
assert erf2(x, y).diff(y) == erf2(x, y).fdiff(2)
assert erf2(x, y).diff(x) == -2*exp(-x**2)/sqrt(pi)
assert erf2(x, y).diff(y) == 2*exp(-y**2)/sqrt(pi)
raises(ArgumentIndexError, lambda: erf2(x, y).fdiff(3))
assert erf2(x, y).is_extended_real is None
xr, yr = symbols('xr yr', extended_real=True)
assert erf2(xr, yr).is_extended_real is True
def test_order_conjugate_transpose():
x = Symbol('x', real=True)
y = Symbol('y', imaginary=True)
assert conjugate(Order(x)) == Order(conjugate(x))
assert conjugate(Order(y)) == Order(conjugate(y))
assert conjugate(Order(x**2)) == Order(conjugate(x)**2)
assert conjugate(Order(y**2)) == Order(conjugate(y)**2)
assert transpose(Order(x)) == Order(transpose(x))
assert transpose(Order(y)) == Order(transpose(y))
assert transpose(Order(x**2)) == Order(transpose(x)**2)
assert transpose(Order(y**2)) == Order(transpose(y)**2)
def test_assoc_laguerre():
n = Symbol("n")
m = Symbol("m")
alpha = Symbol("alpha")
# generalized Laguerre polynomials:
assert assoc_laguerre(0, alpha, x) == 1
assert assoc_laguerre(1, alpha, x) == -x + alpha + 1
assert assoc_laguerre(2, alpha, x).expand() == \
(x**2/2 - (alpha + 2)*x + (alpha + 2)*(alpha + 1)/2).expand()
assert assoc_laguerre(3, alpha, x).expand() == \
(-x**3/6 + (alpha + 3)*x**2/2 - (alpha + 2)*(alpha + 3)*x/2 +
(alpha + 1)*(alpha + 2)*(alpha + 3)/6).expand()
# Test the lowest 10 polynomials with laguerre_poly, to make sure it works:
for i in range(10):
assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
X = assoc_laguerre(n, m, x)
assert isinstance(X, assoc_laguerre)
assert assoc_laguerre(n, 0, x) == laguerre(n, x)
assert assoc_laguerre(n, alpha, 0) == binomial(alpha + n, alpha)
p = Symbol("p", positive=True)
assert assoc_laguerre(p, alpha, oo) == (-1)**p * oo
assert assoc_laguerre(p, alpha, -oo) is oo
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
_k = Dummy('k')
assert diff(assoc_laguerre(n, alpha, x), alpha).dummy_eq(
Sum(assoc_laguerre(_k, alpha, x) / (-alpha + n), (_k, 0, n - 1)))
assert conjugate(assoc_laguerre(n, alpha, x)) == \
assoc_laguerre(n, conjugate(alpha), conjugate(x))
assert assoc_laguerre(n, alpha, x).rewrite('polynomial').dummy_eq(
gamma(alpha + n + 1) * Sum(
x**_k * RisingFactorial(-n, _k) /
(factorial(_k) * gamma(_k + alpha + 1)),
(_k, 0, n)) / factorial(n))
raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, x))
raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(1))
raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(4))
def test_complexfunctions():
xt, yt = aesara_code(x, dtypes={x: 'complex128'
}), aesara_code(y,
dtypes={y: 'complex128'})
from sympy.functions.elementary.complexes import conjugate
from aesara.tensor import as_tensor_variable as atv
from aesara.tensor import complex as cplx
assert theq(aesara_code(y * conjugate(x)), yt * (xt.conj()))
assert theq(aesara_code((1 + 2j) * x),
xt * (atv(1.0) + atv(2.0) * cplx(0, 1)))
def test_li():
z = Symbol("z")
zr = Symbol("z", real=True)
zp = Symbol("z", positive=True)
zn = Symbol("z", negative=True)
assert li(0) is S.Zero
assert li(1) is -oo
assert li(oo) is oo
assert isinstance(li(z), li)
assert unchanged(li, -zp)
assert unchanged(li, zn)
assert diff(li(z), z) == 1/log(z)
assert conjugate(li(z)) == li(conjugate(z))
assert conjugate(li(-zr)) == li(-zr)
assert unchanged(conjugate, li(-zp))
assert unchanged(conjugate, li(zn))
assert li(z).rewrite(Li) == Li(z) + li(2)
assert li(z).rewrite(Ei) == Ei(log(z))
assert li(z).rewrite(uppergamma) == (-log(1/log(z))/2 - log(-log(z)) +
log(log(z))/2 - expint(1, -log(z)))
assert li(z).rewrite(Si) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Ci) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Shi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(Chi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(hyper) ==(log(z)*hyper((1, 1), (2, 2), log(z)) -
log(1/log(z))/2 + log(log(z))/2 + EulerGamma)
assert li(z).rewrite(meijerg) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 -
meijerg(((), (1,)), ((0, 0), ()), -log(z)))
assert gruntz(1/li(z), z, oo) is S.Zero
assert li(z).series(z) == log(z)**5/600 + log(z)**4/96 + log(z)**3/18 + log(z)**2/4 + \
log(z) + log(log(z)) + EulerGamma
raises(ArgumentIndexError, lambda: li(z).fdiff(2))
def _check(roots):
# this is the desired invariant for roots returned
# by all_roots. It is trivially true for linear
# polynomials.
nreal = sum([1 if i.is_real else 0 for i in roots])
assert list(sorted(roots[:nreal])) == list(roots[:nreal])
for ix in range(nreal, len(roots), 2):
if not (roots[ix + 1] == roots[ix]
or roots[ix + 1] == conjugate(roots[ix])):
return False
return True
def test_complexfunctions():
with warns_deprecated_sympy():
xt, yt = theano_code_(x, dtypes={x: 'complex128'}), theano_code_(
y, dtypes={y: 'complex128'})
from sympy.functions.elementary.complexes import conjugate
from theano.tensor import as_tensor_variable as atv
from theano.tensor import complex as cplx
with warns_deprecated_sympy():
assert theq(theano_code_(y * conjugate(x)), yt * (xt.conj()))
assert theq(theano_code_((1 + 2j) * x),
xt * (atv(1.0) + atv(2.0) * cplx(0, 1)))
def measure_all(qubit, format='sympy', normalize=True):
"""Perform an ensemble measurement of all qubits.
Parameters
==========
qubit : Qubit, Add
The qubit to measure. This can be any Qubit or a linear combination
of them.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
=======
result : list
A list that consists of primitive states and their probabilities.
Examples
========
>>> from sympy.physics.quantum.qubit import Qubit, measure_all
>>> from sympy.physics.quantum.gate import H
>>> from sympy.physics.quantum.qapply import qapply
>>> c = H(0)*H(1)*Qubit('00')
>>> c
H(0)*H(1)*|00>
>>> q = qapply(c)
>>> measure_all(q)
[(|00>, 1/4), (|01>, 1/4), (|10>, 1/4), (|11>, 1/4)]
"""
m = qubit_to_matrix(qubit, format)
if format == 'sympy':
results = []
if normalize:
m = m.normalized()
size = max(m.shape) # Max of shape to account for bra or ket
nqubits = int(math.log(size) / math.log(2))
for i in range(size):
if m[i] != 0.0:
results.append(
(Qubit(IntQubit(i,
nqubits=nqubits)), m[i] * conjugate(m[i])))
return results
else:
raise NotImplementedError(
"This function cannot handle non-SymPy matrix formats yet")
def test_laguerre():
n = Symbol("n")
m = Symbol("m", negative=True)
# Laguerre polynomials:
assert laguerre(0, x) == 1
assert laguerre(1, x) == -x + 1
assert laguerre(2, x) == x**2 / 2 - 2 * x + 1
assert laguerre(3, x) == -x**3 / 6 + 3 * x**2 / 2 - 3 * x + 1
assert laguerre(-2, x) == (x + 1) * exp(x)
X = laguerre(n, x)
assert isinstance(X, laguerre)
assert laguerre(n, 0) == 1
assert laguerre(n, oo) == (-1)**n * oo
assert laguerre(n, -oo) is oo
assert conjugate(laguerre(n, x)) == laguerre(n, conjugate(x))
_k = Dummy('k')
assert laguerre(n, x).rewrite("polynomial").dummy_eq(
Sum(x**_k * RisingFactorial(-n, _k) / factorial(_k)**2, (_k, 0, n)))
assert laguerre(m, x).rewrite("polynomial").dummy_eq(
exp(x) * Sum((-x)**_k * RisingFactorial(m + 1, _k) / factorial(_k)**2,
(_k, 0, -m - 1)))
assert diff(laguerre(n, x), x) == -assoc_laguerre(n - 1, 1, x)
k = Symbol('k')
assert laguerre(-n, x) == exp(x) * laguerre(n - 1, -x)
assert laguerre(-3, x) == exp(x) * laguerre(2, -x)
assert unchanged(laguerre, -n + k, x)
raises(ValueError, lambda: laguerre(-2.1, x))
raises(ValueError, lambda: laguerre(Rational(5, 2), x))
raises(ArgumentIndexError, lambda: laguerre(n, x).fdiff(1))
raises(ArgumentIndexError, lambda: laguerre(n, x).fdiff(3))
def _(mat, assumptions):
rows, cols = mat.shape
ret_val = True
for i in range(rows):
for j in range(i, cols):
cond = fuzzy_bool(Eq(mat[i, j], -conjugate(mat[j, i])))
if cond is None:
ret_val = None
if cond == False:
return False
if ret_val is None:
raise MDNotImplementedError
return ret_val
def test_betainc_regularized():
a, b, x1, x2 = symbols('a b x1 x2')
assert unchanged(betainc_regularized, a, b, x1, x2)
assert unchanged(betainc_regularized, a, b, 0, x1)
assert betainc_regularized(3, 5, 0, -1).is_real == True
assert betainc_regularized(3, 5, 0, x2).is_real is None
assert conjugate(betainc_regularized(3*I, 1, 2 + I, 1 + 2*I)) == betainc_regularized(-3*I, 1, 2 - I, 1 - 2*I)
assert betainc_regularized(a, b, 0, 1).rewrite(Integral) == 1
assert betainc_regularized(1, 2, x1, x2).rewrite(hyper) == 2*x2*hyper((1, -1), (2,), x2) - 2*x1*hyper((1, -1), (2,), x1)
assert betainc_regularized(4, 1, 5, 5).evalf() == 0
def test_betainc():
a, b, x1, x2 = symbols('a b x1 x2')
assert unchanged(betainc, a, b, x1, x2)
assert unchanged(betainc, a, b, 0, x1)
assert betainc(1, 2, 0, -5).is_real == True
assert betainc(1, 2, 0, x2).is_real is None
assert conjugate(betainc(I, 2, 3 - I, 1 + 4*I)) == betainc(-I, 2, 3 + I, 1 - 4*I)
assert betainc(a, b, 0, 1).rewrite(Integral).dummy_eq(beta(a, b).rewrite(Integral))
assert betainc(1, 2, 0, x2).rewrite(hyper) == x2*hyper((1, -1), (2,), x2)
assert betainc(1, 2, 3, 3).evalf() == 0
def test_beta():
x, y = symbols('x y')
t = Dummy('t')
assert unchanged(beta, x, y)
assert beta(5, -3).is_real == True
assert beta(3, y).is_real is None
assert expand_func(beta(x, y)) == gamma(x)*gamma(y)/gamma(x + y)
assert expand_func(beta(x, y) - beta(y, x)) == 0 # Symmetric
assert expand_func(beta(x, y)) == expand_func(beta(x, y + 1) + beta(x + 1, y)).simplify()
assert diff(beta(x, y), x) == beta(x, y)*(polygamma(0, x) - polygamma(0, x + y))
assert diff(beta(x, y), y) == beta(x, y)*(polygamma(0, y) - polygamma(0, x + y))
assert conjugate(beta(x, y)) == beta(conjugate(x), conjugate(y))
raises(ArgumentIndexError, lambda: beta(x, y).fdiff(3))
assert beta(x, y).rewrite(gamma) == gamma(x)*gamma(y)/gamma(x + y)
assert beta(x).rewrite(gamma) == gamma(x)**2/gamma(2*x)
assert beta(x, y).rewrite(Integral).dummy_eq(Integral(t**(x - 1) * (1 - t)**(y - 1), (t, 0, 1)))
def _entry(self, i, j):
return conjugate(self.arg._entry(j, i))
def test_sympy__functions__elementary__complexes__conjugate():
from sympy.functions.elementary.complexes import conjugate
assert _test_args(conjugate(x))
def _eval_trace(self):
from trace import Trace
return conjugate(Trace(self.arg))
def _eval_conjugate(self):
from sympy.functions.elementary.complexes import conjugate
return Piecewise(*[(conjugate(e), c) for e, c in self.args])
