def test_hankel_transform():
from sympy import sinh, cosh, gamma, sqrt, exp
r = Symbol("r")
k = Symbol("k")
nu = Symbol("nu")
m = Symbol("m")
a = symbols("a")
assert hankel_transform(1/r, r, k, 0) == 1/k
assert inverse_hankel_transform(1/k, k, r, 0) == 1/r
assert hankel_transform(
1/r**m, r, k, 0) == 2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2)
assert inverse_hankel_transform(
2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2), k, r, 0) == r**(-m)
assert hankel_transform(1/r**m, r, k, nu) == (
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2))
assert inverse_hankel_transform(2**(-m + 1)*k**(
m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2), k, r, nu) == r**(-m)
assert hankel_transform(r**nu*exp(-a*r), r, k, nu) == \
2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(
3)/2)*gamma(nu + S(3)/2)/sqrt(pi)
assert inverse_hankel_transform(
2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(3)/2)*gamma(
nu + S(3)/2)/sqrt(pi), k, r, nu) == r**nu*exp(-a*r)
def test_evalf_fast_series():
# Euler transformed series for sqrt(1+x)
assert NS(Sum(
fac(2*n + 1)/fac(n)**2/2**(3*n + 1), (n, 0, oo)), 100) == NS(sqrt(2), 100)
# Some series for exp(1)
estr = NS(E, 100)
assert NS(Sum(1/fac(n), (n, 0, oo)), 100) == estr
assert NS(1/Sum((1 - 2*n)/fac(2*n), (n, 0, oo)), 100) == estr
assert NS(Sum((2*n + 1)/fac(2*n), (n, 0, oo)), 100) == estr
assert NS(Sum((4*n + 3)/2**(2*n + 1)/fac(2*n + 1), (n, 0, oo))**2, 100) == estr
pistr = NS(pi, 100)
# Ramanujan series for pi
assert NS(9801/sqrt(8)/Sum(fac(
4*n)*(1103 + 26390*n)/fac(n)**4/396**(4*n), (n, 0, oo)), 100) == pistr
assert NS(1/Sum(
binomial(2*n, n)**3 * (42*n + 5)/2**(12*n + 4), (n, 0, oo)), 100) == pistr
# Machin's formula for pi
assert NS(16*Sum((-1)**n/(2*n + 1)/5**(2*n + 1), (n, 0, oo)) -
4*Sum((-1)**n/(2*n + 1)/239**(2*n + 1), (n, 0, oo)), 100) == pistr
# Apery's constant
astr = NS(zeta(3), 100)
P = 126392*n**5 + 412708*n**4 + 531578*n**3 + 336367*n**2 + 104000* \
n + 12463
assert NS(Sum((-1)**n * P / 24 * (fac(2*n + 1)*fac(2*n)*fac(
n))**3 / fac(3*n + 2) / fac(4*n + 3)**3, (n, 0, oo)), 100) == astr
assert NS(Sum((-1)**n * (205*n**2 + 250*n + 77)/64 * fac(n)**10 /
fac(2*n + 1)**5, (n, 0, oo)), 100) == astr
def test_twave():
A1, phi1, A2, phi2, f = symbols('A1, phi1, A2, phi2, f')
n = Symbol('n') # Refractive index
t = Symbol('t') # Time
x = Symbol('x') # Spatial varaible
k = Symbol('k') # Wave number
E = Function('E')
w1 = TWave(A1, f, phi1)
w2 = TWave(A2, f, phi2)
assert w1.amplitude == A1
assert w1.frequency == f
assert w1.phase == phi1
assert w1.wavelength == c/(f*n)
assert w1.time_period == 1/f
w3 = w1 + w2
assert w3.amplitude == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2)
assert w3.frequency == f
assert w3.wavelength == c/(f*n)
assert w3.time_period == 1/f
assert w3.angular_velocity == 2*pi*f
assert w3.wavenumber == 2*pi*f*n/c
assert simplify(w3.rewrite('sin') - sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*sin(pi*f*n*x*s/(149896229*m) - 2*pi*f*t + atan2(A1*cos(phi1)
+ A2*cos(phi2), A1*sin(phi1) + A2*sin(phi2)) + pi/2)) == 0
assert w3.rewrite('pde') == epsilon*mu*Derivative(E(x, t), t, t) + Derivative(E(x, t), x, x)
assert w3.rewrite(cos) == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*cos(pi*f*n*x*s/(149896229*m) - 2*pi*f*t + atan2(A1*cos(phi1)
+ A2*cos(phi2), A1*sin(phi1) + A2*sin(phi2)))
assert w3.rewrite('exp') == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*exp(I*(pi*f*n*x*s/(149896229*m) - 2*pi*f*t
+ atan2(A1*cos(phi1) + A2*cos(phi2), A1*sin(phi1) + A2*sin(phi2))))
def root_mul_rule(integral):
integrand, symbol = integral
a = sympy.Wild("a", exclude=[symbol])
b = sympy.Wild("b", exclude=[symbol])
c = sympy.Wild("c")
match = integrand.match(sympy.sqrt(a * symbol + b) * c)
if not match:
return
a, b, c = match[a], match[b], match[c]
d = sympy.Wild("d", exclude=[symbol])
e = sympy.Wild("e", exclude=[symbol])
f = sympy.Wild("f")
recursion_test = c.match(sympy.sqrt(d * symbol + e) * f)
if recursion_test:
return
u = sympy.Dummy("u")
u_func = sympy.sqrt(a * symbol + b)
integrand = integrand.subs(u_func, u)
integrand = integrand.subs(symbol, (u ** 2 - b) / a)
integrand = integrand * 2 * u / a
next_step = integral_steps(integrand, u)
if next_step:
return URule(u, u_func, None, next_step, integrand, symbol)
def test_director_circle():
x, y, a, b = symbols('x y a b')
e = Ellipse((x, y), a, b)
# the general result
assert e.director_circle() == Circle((x, y), sqrt(a**2 + b**2))
# a special case where Ellipse is a Circle
assert Circle((3, 4), 8).director_circle() == Circle((3, 4), 8*sqrt(2))
def make_inverse_trig(RuleClass, base_exp, a, sign_a, b, sign_b):
u_var = sympy.Dummy("u")
current_base = base
current_symbol = symbol
constant = u_func = u_constant = substep = None
factored = integrand
if a != 1:
constant = a**base_exp
current_base = sign_a + sign_b * (b/a) * current_symbol**2
factored = current_base ** base_exp
if (b/a) != 1:
u_func = sympy.sqrt(b/a) * symbol
u_constant = sympy.sqrt(a/b)
current_symbol = u_var
current_base = sign_a + sign_b * current_symbol**2
substep = RuleClass(current_base ** base_exp, current_symbol)
if u_func is not None:
if u_constant != 1:
substep = ConstantTimesRule(
u_constant, current_base ** base_exp, substep,
u_constant * current_base ** base_exp, symbol)
substep = URule(u_var, u_func, u_constant, substep, factored, symbol)
if constant is not None:
substep = ConstantTimesRule(constant, factored, substep, integrand, symbol)
return substep
def test_pretty_functions():
f = Function('f')
# Simple
assert pretty( (2*x + exp(x)) ) in [' x \ne + 2*x', ' x\n2*x + e ']
assert pretty(abs(x)) == '|x|'
assert pretty(abs(x/(x**2+1))) in [
'| x |\n|------|\n| 2|\n|1 + x |',
'| x |\n|------|\n| 2 |\n|x + 1|']
assert pretty(conjugate(x)) == '_\nx'
assert pretty(conjugate(f(x+1))) in [
'________\nf(1 + x)',
'________\nf(x + 1)']
# Univariate/Multivariate functions
assert pretty(f(x)) == 'f(x)'
assert pretty(f(x, y)) == 'f(x, y)'
assert pretty(f(x/(y+1), y)) in [
' / x \\\nf|-----, y|\n \\1 + y /',
' / x \\\nf|-----, y|\n \\y + 1 /',
]
# Nesting of square roots
assert pretty( sqrt((sqrt(x+1))+1) ) in [
' _______________\n / _______ \n\\/ 1 + \\/ 1 + x ',
' _______________\n / _______ \n\\/ \\/ x + 1 + 1 ']
# Function powers
assert pretty( sin(x)**2 ) == ' 2 \nsin (x)'
# Conjugates
a,b = map(Symbol, 'ab')
def test_evalf_ramanujan():
assert NS(exp(pi*sqrt(163)) - 640320**3 - 744, 10) == '-7.499274028e-13'
# A related identity
A = 262537412640768744*exp(-pi*sqrt(163))
B = 196884*exp(-2*pi*sqrt(163))
C = 103378831900730205293632*exp(-3*pi*sqrt(163))
assert NS(1 - A - B + C, 10) == '1.613679005e-59'
def eval_trigsubstitution(theta, func, rewritten, substep, restriction, integrand, symbol):
func = func.subs(sympy.sec(theta), 1/sympy.cos(theta))
trig_function = list(func.find(TrigonometricFunction))
assert len(trig_function) == 1
trig_function = trig_function[0]
relation = sympy.solve(symbol - func, trig_function)
assert len(relation) == 1
numer, denom = sympy.fraction(relation[0])
if isinstance(trig_function, sympy.sin):
opposite = numer
hypotenuse = denom
adjacent = sympy.sqrt(denom**2 - numer**2)
inverse = sympy.asin(relation[0])
elif isinstance(trig_function, sympy.cos):
adjacent = numer
hypotenuse = denom
opposite = sympy.sqrt(denom**2 - numer**2)
inverse = sympy.acos(relation[0])
elif isinstance(trig_function, sympy.tan):
opposite = numer
adjacent = denom
hypotenuse = sympy.sqrt(denom**2 + numer**2)
inverse = sympy.atan(relation[0])
substitution = [
(sympy.sin(theta), opposite/hypotenuse),
(sympy.cos(theta), adjacent/hypotenuse),
(sympy.tan(theta), opposite/adjacent),
(theta, inverse)
]
return sympy.Piecewise(
(_manualintegrate(substep).subs(substitution).trigsimp(), restriction)
)
def test_solve_complex_sqrt():
assert solveset_complex(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S(1), S(2))
assert solveset_complex(sqrt(5*x + 6) - (2 + 2*I) - x, x) == \
FiniteSet(-S(2), 3 - 4*I)
assert solveset_complex(4*x*(1 - a * sqrt(x)), x) == \
FiniteSet(S(0), 1 / a ** 2)
def test_as_ordered_terms():
f, g = symbols("f,g", cls=Function)
assert x.as_ordered_terms() == [x]
assert (sin(x) ** 2 * cos(x) + sin(x) * cos(x) ** 2 + 1).as_ordered_terms() == [
sin(x) ** 2 * cos(x),
sin(x) * cos(x) ** 2,
1,
]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Add(*args)
assert expr.as_ordered_terms() == args
assert (1 + 4 * sqrt(3) * pi * x).as_ordered_terms() == [4 * pi * x * sqrt(3), 1]
assert (2 + 3 * I).as_ordered_terms() == [2, 3 * I]
assert (-2 + 3 * I).as_ordered_terms() == [-2, 3 * I]
assert (2 - 3 * I).as_ordered_terms() == [2, -3 * I]
assert (-2 - 3 * I).as_ordered_terms() == [-2, -3 * I]
assert (4 + 3 * I).as_ordered_terms() == [4, 3 * I]
assert (-4 + 3 * I).as_ordered_terms() == [-4, 3 * I]
assert (4 - 3 * I).as_ordered_terms() == [4, -3 * I]
assert (-4 - 3 * I).as_ordered_terms() == [-4, -3 * I]
f = x ** 2 * y ** 2 + x * y ** 4 + y + 2
assert f.as_ordered_terms(order="lex") == [x ** 2 * y ** 2, x * y ** 4, y, 2]
assert f.as_ordered_terms(order="grlex") == [x * y ** 4, x ** 2 * y ** 2, y, 2]
assert f.as_ordered_terms(order="rev-lex") == [2, y, x * y ** 4, x ** 2 * y ** 2]
assert f.as_ordered_terms(order="rev-grlex") == [2, y, x ** 2 * y ** 2, x * y ** 4]
def square_root_of_expr(expr):
"""
If expression is product of even powers then every power is divided
by two and the product is returned. If some terms in product are
not even powers the sqrt of the absolute value of the expression is
returned. If the expression is a number the sqrt of the absolute
value of the number is returned.
"""
if expr.is_number:
if expr > 0:
return(sqrt(expr))
else:
return(sqrt(-expr))
else:
expr = trigsimp(expr)
(coef, pow_lst) = sqf_list(expr)
if coef != S(1):
if coef.is_number:
coef = square_root_of_expr(coef)
else:
coef = sqrt(abs(coef)) # Product coefficient not a number
for p in pow_lst:
(f, n) = p
if n % 2 != 0:
return(sqrt(abs(expr))) # Product not all even powers
else:
coef *= f ** (n / 2) # Positive sqrt of the square of an expression
return coef
def test_uselogcombine_1():
assert solveset_real(log(x - 3) + log(x + 3), x) == \
FiniteSet(sqrt(10))
assert solveset_real(log(x + 1) - log(2*x - 1), x) == FiniteSet(2)
assert solveset_real(log(x + 3) + log(1 + 3/x) - 3) == FiniteSet(
-3 + sqrt(-12 + exp(3))*exp(S(3)/2)/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(S(3)/2)/2 - 3 + exp(3)/2)
def test_guess_rational_cv():
# rational functions
assert guess_solve_strategy( (x+1)/(x**2 + 2), x) #== GS_RATIONAL
assert guess_solve_strategy( (x - y**3)/(y**2*sqrt(1 - y**2)), y) #== GS_RATIONAL_CV_1
# rational functions via the change of variable y -> x**n
assert guess_solve_strategy( (sqrt(x) + 1)/(x**Rational(1,3) + sqrt(x) + 1), x ) \
def test_issue_1364():
assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
assert solve(x**x) == []
assert solve(x**x - 2) == [exp(LambertW(log(2)))]
assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
assert solve((a/x + exp(x/2)).diff(x), x) == [4*LambertW(sqrt(2)*sqrt(a)/4)]
def test_solve_polynomial1():
assert solve(3*x-2, x) == [Rational(2,3)]
assert solve(Eq(3*x, 2), x) == [Rational(2,3)]
assert solve(x**2-1, x) in [[-1, 1], [1, -1]]
assert solve(Eq(x**2, 1), x) in [[-1, 1], [1, -1]]
assert solve( x - y**3, x) == [y**3]
assert sorted(solve( x - y**3, y)) == sorted([
(-x**Rational(1,3))/2 + I*sqrt(3)*x**Rational(1,3)/2,
x**Rational(1,3),
(-x**Rational(1,3))/2 - I*sqrt(3)*x**Rational(1,3)/2,
])
a11,a12,a21,a22,b1,b2 = symbols('a11,a12,a21,a22,b1,b2')
assert solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y) == \
{
x : (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y : (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
solution = {y: S.Zero, x: S.Zero}
assert solve((x - y, x+y), x, y ) == solution
assert solve((x - y, x+y), (x, y)) == solution
assert solve((x - y, x+y), [x, y]) == solution
assert set(solve(x**3 - 15*x - 4, x)) == set([-2 + 3**Rational(1,2),
S(4),
-2 - 3**Rational(1,2) ])
assert sorted(solve((x**2 - 1)**2 - a, x)) == \
sorted([sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a))])
def test_checking():
assert set(solve(x*(x - y/x),x, check=False)) == set([sqrt(y), S(0), -sqrt(y)])
assert set(solve(x*(x - y/x),x, check=True)) == set([sqrt(y), -sqrt(y)])
# {x: 0, y: 4} sets denominator to 0 in the following so system should return None
assert solve((1/(1/x + 2), 1/(y - 3) - 1)) is None
# 0 sets denominator of 1/x to zero so [] is returned
assert solve(1/(1/x + 2)) == []
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_polarify():
from sympy import polar_lift, polarify
x = Symbol('x')
z = Symbol('z', polar=True)
f = Function('f')
ES = {}
assert polarify(-1) == (polar_lift(-1), ES)
assert polarify(1 + I) == (polar_lift(1 + I), ES)
assert polarify(exp(x), subs=False) == exp(x)
assert polarify(1 + x, subs=False) == 1 + x
assert polarify(f(I) + x, subs=False) == f(polar_lift(I)) + x
assert polarify(x, lift=True) == polar_lift(x)
assert polarify(z, lift=True) == z
assert polarify(f(x), lift=True) == f(polar_lift(x))
assert polarify(1 + x, lift=True) == polar_lift(1 + x)
assert polarify(1 + f(x), lift=True) == polar_lift(1 + f(polar_lift(x)))
newex, subs = polarify(f(x) + z)
assert newex.subs(subs) == f(x) + z
mu = Symbol("mu")
sigma = Symbol("sigma", positive=True)
# Make sure polarify(lift=True) doesn't try to lift the integration
# variable
assert polarify(
Integral(sqrt(2)*x*exp(-(-mu + x)**2/(2*sigma**2))/(2*sqrt(pi)*sigma),
(x, -oo, oo)), lift=True) == Integral(sqrt(2)*(sigma*exp_polar(0))**exp_polar(I*pi)*
exp((sigma*exp_polar(0))**(2*exp_polar(I*pi))*exp_polar(I*pi)*polar_lift(-mu + x)**
(2*exp_polar(0))/2)*exp_polar(0)*polar_lift(x)/(2*sqrt(pi)), (x, -oo, oo))
def test_factorial2_rewrite():
n = Symbol('n', integer=True)
assert factorial2(n).rewrite(gamma) == \
2**(n/2)*Piecewise((1, Eq(Mod(n, 2), 0)), (sqrt(2)/sqrt(pi), Eq(Mod(n, 2), 1)))*gamma(n/2 + 1)
assert factorial2(2*n).rewrite(gamma) == 2**n*gamma(n + 1)
assert factorial2(2*n + 1).rewrite(gamma) == \
sqrt(2)*2**(n + 1/2)*gamma(n + 3/2)/sqrt(pi)
def test_polysys():
from sympy.abc import x, y
assert solve([x**2 + 2/y - 2 , x + y - 3], [x, y]) == \
[(1, 2), (1 + sqrt(5), 2 - sqrt(5)), (1 - sqrt(5), 2 + sqrt(5))]
assert solve([x**2 + y - 2, x**2 + y]) is None
# the ordering should be whatever the user requested
assert solve([x**2 + y - 3, x - y - 4], (x, y)) != solve([x**2 + y - 3, x - y - 4], (y, x))
def test_nthroot1():
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**20
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**30
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
def eta_fil(self, x, V_app, apprx=(0, 0, 0, 0)):
m_eff = self.m_r * const.electron_mass
mpmath.mp.dps = 20
x0 = Symbol('x0') # eta_fil
x1 = Symbol('x1') # eta_ac
x2 = Symbol('x2') # eta_hop
x3 = Symbol('x3') # V_tunnel
f0 = const.Boltzmann * self.T / (1 - self.alpha) / const.elementary_charge / self.z * \
ln(self.A_fil/self.A_ac*(exp(- self.alpha * const.elementary_charge * self.z / const.Boltzmann / self.T * x0) - 1) + 1) - x1# eta_ac = f(eta_fil) x1 = f(x0)
f1 = x*2*const.Boltzmann*self.T/self.a/self.z/const.elementary_charge*\
asinh(self.j_0et/self.j_0hop*(exp(- self.alpha * const.elementary_charge * self.z / const.Boltzmann / self.T * x0) - 1)) - x2# eta_hop = f(eta_fil)
f2 = x1 - x0 + x2 - x3
f3 = -V_app + ((self.C * 3 * sqrt(2 * m_eff * ((4+x3/2)*const.elementary_charge)) / 2 / x * (const.elementary_charge / const.Planck)**2 * \
exp(- 4 * const.pi * x / const.Planck * sqrt(2 * m_eff * ((4+x3/2)*const.elementary_charge))) * self.A_fil*x3)
+ (self.j_0et*self.A_fil*(exp(-self.alpha*const.elementary_charge*self.z*x0/const.Boltzmann/self.T) - 1))) * (self.R_el + self.R_S + self.rho_fil*(self.L - x) / self.A_fil) \
+ x3
eta_fil, eta_ac, eta_hop, V_tunnel = nsolve((f0, f1, f2, f3), [x0, x1, x2, x3], apprx)
eta_fil = np.real(np.complex128(eta_fil))
eta_ac = np.real(np.complex128(eta_ac))
eta_hop = np.real(np.complex128(eta_hop))
V_tunnel = np.real(np.complex128(V_tunnel))
current = ((self.C * 3 * sqrt(2 * m_eff * ((4+V_tunnel)*const.elementary_charge)) / 2 / x * (const.elementary_charge / const.Planck)**2 * \
exp(- 4 * const.pi * x / const.Planck * sqrt(2 * m_eff * ((4+V_tunnel)*const.elementary_charge))) * self.A_fil*V_tunnel)
+ (self.j_0et*self.A_fil*(exp(-self.alpha*const.elementary_charge*self.z*eta_fil/const.Boltzmann/self.T) - 1)))
print(eta_fil, eta_ac, eta_hop, V_tunnel)
# print(eta_ac - eta_fil + eta_hop - V_tunnel)
return eta_fil, eta_ac, eta_hop, V_tunnel, current
def test__erfs():
assert _erfs(z).diff(z) == -2/sqrt(S.Pi)+2*z*_erfs(z)
assert _erfs(1/z).series(z) == z/sqrt(pi) - z**3/(2*sqrt(pi)) + 3*z**5/(4*sqrt(pi)) + O(z**6)
assert expand(erf(z).rewrite('tractable').diff(z).rewrite('intractable')) == erf(z).diff(z)
assert _erfs(z).rewrite("intractable") == (-erf(z) + 1)*exp(z**2)
def test_roots_quartic():
assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0]
assert roots_quartic(Poly(x**4 + x**3, x)) in [
[-1,0,0,0],
[0,-1,0,0],
[0,0,-1,0],
[0,0,0,-1]
]
assert roots_quartic(Poly(x**4 - x**3, x)) in [
[1,0,0,0],
[0,1,0,0],
[0,0,1,0],
[0,0,0,1]
]
lhs = roots_quartic(Poly(x**4 + x, x))
rhs = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
# test of all branches of roots quartic
for i, (a, b, c, d) in enumerate([(1, 2, 3, 0),
(3, -7, -9, 9),
(1, 2, 3, 4),
(1, 2, 3, 4),
(-7, -3, 3, -6),
(-3, 5, -6, -4)]):
if i == 2:
c = -a*(a**2/S(8) - b/S(2))
elif i == 3:
d = a*(a*(3*a**2/S(256) - b/S(16)) + c/S(4))
eq = x**4 + a*x**3 + b*x**2 + c*x + d
ans = roots_quartic(Poly(eq, x))
assert all([eq.subs(x, ai).n(chop=True) == 0 for ai in ans])
def test_rayleigh():
sigma = Symbol("sigma", positive=True)
X = Rayleigh('x', sigma)
assert density(X)(x) == x*exp(-x**2/(2*sigma**2))/sigma**2
assert E(X) == sqrt(2)*sqrt(pi)*sigma/2
assert variance(X) == -pi*sigma**2/2 + 2*sigma**2
def test_expr_sorting():
f, g = symbols('f,g', cls=Function)
exprs = [1/x**2, 1/x, sqrt(sqrt(x)), sqrt(x), x, sqrt(x)**3, x**2]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x, 2*x, 2*x**2, 2*x**3, x**n, 2*x**n, sin(x), sin(x)**n, sin(x**2), cos(x), cos(x**2), tan(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x + 1, x**2 + x + 1, x**3 + x**2 + x + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [S(4), x - 3*I/2, x + 3*I/2, x - 4*I + 1, x + 4*I + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(x), g(x), exp(x), sin(x), cos(x), factorial(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [Tuple(x, y), Tuple(x, z), Tuple(x, y, z)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[3], [1, 2]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [1, 2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{x: -y}, {x: y}]
assert sorted(exprs, key=default_sort_key) == exprs
def test_cse_single2():
# Simple substitution, test for being able to pass the expression directly
e = Add(Pow(x+y,2), sqrt(x+y))
substs, reduced = cse(e, optimizations=[])
assert substs == [(x0, x+y)]
assert reduced == [sqrt(x0) + x0**2]
assert isinstance(cse(Matrix([[1]]))[1][0], Matrix)
def test_lognormal():
mean = Symbol('mu', real=True, finite=True)
std = Symbol('sigma', positive=True, real=True, finite=True)
X = LogNormal('x', mean, std)
# The sympy integrator can't do this too well
#assert E(X) == exp(mean+std**2/2)
#assert variance(X) == (exp(std**2)-1) * exp(2*mean + std**2)
# Right now, only density function and sampling works
# Test sampling: Only e^mean in sample std of 0
for i in range(3):
X = LogNormal('x', i, 0)
assert S(sample(X)) == N(exp(i))
# The sympy integrator can't do this too well
#assert E(X) ==
mu = Symbol("mu", real=True)
sigma = Symbol("sigma", positive=True)
X = LogNormal('x', mu, sigma)
assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2
/(2*sigma**2))/(2*x*sqrt(pi)*sigma))
X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1
assert density(X)(x) == sqrt(2)*exp(-log(x)**2/2)/(2*x*sqrt(pi))
def test_logcombine_1():
x, y = symbols("x,y")
a = Symbol("a")
z, w = symbols("z,w", positive=True)
b = Symbol("b", real=True)
assert logcombine(log(x) + 2*log(y)) == log(x) + 2*log(y)
assert logcombine(log(x) + 2*log(y), force=True) == log(x*y**2)
assert logcombine(a*log(w) + log(z)) == a*log(w) + log(z)
assert logcombine(b*log(z) + b*log(x)) == log(z**b) + b*log(x)
assert logcombine(b*log(z) - log(w)) == log(z**b/w)
assert logcombine(log(x)*log(z)) == log(x)*log(z)
assert logcombine(log(w)*log(x)) == log(w)*log(x)
assert logcombine(cos(-2*log(z) + b*log(w))) in [cos(log(w**b/z**2)),
cos(log(z**2/w**b))]
assert logcombine(log(log(x) - log(y)) - log(z), force=True) == \
log(log(x/y)/z)
assert logcombine((2 + I)*log(x), force=True) == (2 + I)*log(x)
assert logcombine((x**2 + log(x) - log(y))/(x*y), force=True) == \
(x**2 + log(x/y))/(x*y)
# the following could also give log(z*x**log(y**2)), what we
# are testing is that a canonical result is obtained
assert logcombine(log(x)*2*log(y) + log(z), force=True) == \
log(z*y**log(x**2))
assert logcombine((x*y + sqrt(x**4 + y**4) + log(x) - log(y))/(pi*x**Rational(2, 3)*
sqrt(y)**3), force=True) == (
x*y + sqrt(x**4 + y**4) + log(x/y))/(pi*x**(S(2)/3)*y**(S(3)/2))
assert logcombine(gamma(-log(x/y))*acos(-log(x/y)), force=True) == \
acos(-log(x/y))*gamma(-log(x/y))
assert logcombine(2*log(z)*log(w)*log(x) + log(z) + log(w)) == \
log(z**log(w**2))*log(x) + log(w*z)
assert logcombine(3*log(w) + 3*log(z)) == log(w**3*z**3)
assert logcombine(x*(y + 1) + log(2) + log(3)) == x*(y + 1) + log(6)
assert logcombine((x + y)*log(w) + (-x - y)*log(3)) == (x + y)*log(w/3)
def _eval_rewrite_as_xp(self, *args, **kwargs):
return (Integer(1) /
sqrt(Integer(2) * hbar * m * omega)) * (I * Px + m * omega * X)
def test_issue_2387():
assert not cos(sqrt(0.5 + I)).n().is_Function
def test_issue1037():
assert cosh(asinh(Integer(3) / 2)) == sqrt(Integer(13) / 4)
def test_wignersemicircle():
R = Symbol("R", positive=True)
X = WignerSemicircle('x', R)
assert density(X)(x) == 2 * sqrt(-x**2 + R**2) / (pi * R**2)
assert E(X) == 0
def test_arcsin():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
X = Arcsin('x', a, b)
assert density(X)(x) == 1 / (pi * sqrt((-x + b) * (x - a)))
# standardized moments of an exponential distribution:
expn_moments = [1, 0, 1, 2, 9, 44, 265, 1854, 14833]
x_nom = [24, 37, 0.5]
x_std = [1, 4, 0.5]
x_vm = np.array([norm_moments, norm_moments, expn_moments])
soerp_symbolic(Z, [x1, x2, x3],
x_nom,
x_std,
x_vm,
title='THREE PART ASSEMBLY')
########################
C, H, M, P, t = sym.symbols('C,H,M,P,t')
Q = C * sym.sqrt((520.0 * H * P) / (M * (t + 460.0)))
Q = Q.subs({C: 38.4})
x_nom = [64.0, 16.0, 361.0, 165.0]
x_std = [0.5, 0.1, 2.0, 0.5]
x_vm = np.array([norm_moments, norm_moments, norm_moments, norm_moments])
soerp_symbolic(Q, [H, M, P, t],
x_nom,
x_std,
x_vm,
title='VOLUMETRIC GAS FLOW THROUGH ORIFICE METER')
########################
x = sym.Symbol('x')
def _apply_operator_SHOKet(self, ket):
temp = ket.n - Integer(1)
if ket.n == Integer(0):
return Integer(0)
else:
return sqrt(ket.n) * SHOKet(temp)
def rayleigh_length(z, r_curv_at_z):
return sp.sqrt(z * (r_curv_at_z - z))
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), x**2, x).doit() == f(sqrt(x))
def _apply_operator_SHOKet(self, ket):
temp = ket.n + Integer(1)
return sqrt(temp) * SHOKet(temp)
def WeylScalars_Cartesian():
# We do not need the barred or hatted quantities calculated when using Cartesian coordinates.
# Instead, we declare the PHYSICAL metric and extrinsic curvature as grid functions.
gammaDD = ixp.register_gridfunctions_for_single_rank2(
"EVOL", "gammaDD", "sym01")
kDD = ixp.register_gridfunctions_for_single_rank2("EVOL", "kDD", "sym01")
tmpgammaUU, detgamma = ixp.symm_matrix_inverter3x3(gammaDD)
detgamma = sp.simplify(detgamma)
gammaUU = ixp.zerorank2()
for i in range(3):
for j in range(3):
gammaUU[i][j] = sp.simplify(tmpgammaUU[i][j])
output_scalars = par.parval_from_str("output_scalars")
global psi4r, psi4i, psi3r, psi3i, psi2r, psi2i, psi1r, psi1i, psi0r, psi0i
# if output_scalars is "all_psis_and_invariants":
# psi4r,psi4i,psi3r,psi3i,psi2r,psi2i,psi1r,psi1i,psi0r,psi0i = sp.symbols("psi4r psi4i\
# psi3r psi3i\
# psi2r psi2i\
# psi1r psi1i\
# psi0r psi0i")
# elif output_scalars is "all_psis":
psi4r,psi4i,psi3r,psi3i,psi2r,psi2i,psi1r,psi1i,psi0r,psi0i = gri.register_gridfunctions("AUX",["psi4r","psi4i",\
"psi3r","psi3i",\
"psi2r","psi2i",\
"psi1r","psi1i",\
"psi0r","psi0i"])
# Step 4.a: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM", DIM)
# Step 4.b: Set the coordinate system to Cartesian
x, y, z = gri.register_gridfunctions("AUX", ["x", "y", "z"])
# Step 5: Set which tetrad is used; at the moment, only one supported option
if par.parval_from_str("WeylScal4NRPy.WeylScalars_Cartesian::TetradChoice"
) == "Approx_QuasiKinnersley":
# Step 5.a: Choose 3 orthogonal vectors. Here, we choose one in the azimuthal
# direction, one in the radial direction, and the cross product of the two.
# Eqs 5.6, 5.7 in https://arxiv.org/pdf/gr-qc/0104063.pdf:
# v_1^a &= [-y,x,0] \\
# v_2^a &= [x,y,z] \\
# v_3^a &= {\rm det}(g)^{1/2} g^{ad} \epsilon_{dbc} v_1^b v_2^c,
v1U = ixp.zerorank1()
v2U = ixp.zerorank1()
v3U = ixp.zerorank1()
v1U[0] = -y
v1U[1] = x
v1U[2] = sp.sympify(0)
v2U[0] = x
v2U[1] = y
v2U[2] = z
LeviCivitaSymbol_rank3 = define_LeviCivitaSymbol_rank3()
for a in range(DIM):
for b in range(DIM):
for c in range(DIM):
for d in range(DIM):
v3U[a] += sp.sqrt(
detgamma) * gammaUU[a][d] * LeviCivitaSymbol_rank3[
d][b][c] * v1U[b] * v2U[c]
for a in range(DIM):
v3U[a] = sp.simplify(v3U[a])
# Step 5.b: Gram-Schmidt orthonormalization of the vectors.
# The w_i^a vectors here are used to temporarily hold values on the way to the final vectors e_i^a
# e_1^a &= \frac{v_1^a}{\omega_{11}} \\
# e_2^a &= \frac{v_2^a - \omega_{12} e_1^a}{\omega_{22}} \\
# e_3^a &= \frac{v_3^a - \omega_{13} e_1^a - \omega_{23} e_2^a}{\omega_{33}}, \\
# Normalize the first vector
w1U = ixp.zerorank1()
for a in range(DIM):
w1U[a] = v1U[a]
omega11 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega11 += w1U[a] * w1U[b] * gammaDD[a][b]
e1U = ixp.zerorank1()
for a in range(DIM):
e1U[a] = w1U[a] / sp.sqrt(omega11)
# Subtract off the portion of the first vector along the second, then normalize
omega12 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega12 += e1U[a] * v2U[b] * gammaDD[a][b]
w2U = ixp.zerorank1()
for a in range(DIM):
w2U[a] = v2U[a] - omega12 * e1U[a]
omega22 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega22 += w2U[a] * w2U[b] * gammaDD[a][b]
e2U = ixp.zerorank1()
for a in range(DIM):
e2U[a] = w2U[a] / sp.sqrt(omega22)
# Subtract off the portion of the first and second vectors along the third, then normalize
omega13 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega13 += e1U[a] * v3U[b] * gammaDD[a][b]
omega23 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega23 += e2U[a] * v3U[b] * gammaDD[a][b]
w3U = ixp.zerorank1()
for a in range(DIM):
w3U[a] = v3U[a] - omega13 * e1U[a] - omega23 * e2U[a]
omega33 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega33 += w3U[a] * w3U[b] * gammaDD[a][b]
e3U = ixp.zerorank1()
for a in range(DIM):
e3U[a] = w3U[a] / sp.sqrt(omega33)
# Step 5.c: Construct the tetrad itself.
# Eqs. 5.6:
# l^a &= \frac{1}{\sqrt{2}} e_2^a \\
# n^a &= -\frac{1}{\sqrt{2}} e_2^a \\
# m^a &= \frac{1}{\sqrt{2}} (e_3^a + i e_1^a) \\
# \overset{*}{m}{}^a &= \frac{1}{\sqrt{2}} (e_3^a - i e_1^a)
isqrt2 = 1 / sp.sqrt(2)
ltetU = ixp.zerorank1()
ntetU = ixp.zerorank1()
#mtetU = ixp.zerorank1()
#mtetccU = ixp.zerorank1()
remtetU = ixp.zerorank1(
) # SymPy does not like trying to take the real/imaginary parts of such a
immtetU = ixp.zerorank1(
) # complicated expression as the Weyl scalars, so we will do it ourselves.
for i in range(DIM):
ltetU[i] = isqrt2 * e2U[i]
ntetU[i] = -isqrt2 * e2U[i]
remtetU[i] = isqrt2 * e3U[i]
immtetU[i] = isqrt2 * e1U[i]
nn = isqrt2
else:
print("Error: TetradChoice == " + par.parval_from_str("TetradChoice") +
" unsupported!")
exit(1)
#Step 6: Declare and construct the second derivative of the metric.
#gammabarDD_dDD = ixp.zerorank4()
#for i in range(DIM):
# for j in range(DIM):
# for k in range(DIM):
# for l in range(DIM):
# gammabarDD_dDD[i][j][k][l] = bssn.hDD_dDD[i][j][k][l]*rfm.ReDD[i][j] + \
# bssn.hDD_dD[i][j][k]*rfm.ReDDdD[i][j][l] + \
# bssn.hDD_dD[i][j][l]*rfm.ReDDdD[i][j][k] + \
# bssn.hDD[i][j]*rfm.ReDDdDD[i][j][k][l] + \
# rfm.ghatDDdDD[i][j][k][l]
gammaDD_dD = ixp.declarerank3("gammaDD_dD", "sym01")
# Define the Christoffel symbols
GammaUDD = ixp.zerorank3(DIM)
for i in range(DIM):
for k in range(DIM):
for l in range(DIM):
for m in range(DIM):
GammaUDD[i][k][l] += (sp.Rational(1,2))*gammaUU[i][m]*\
(gammaDD_dD[m][k][l] + gammaDD_dD[m][l][k] - gammaDD_dD[k][l][m])
# Step 6.a: Declare and construct the Riemann curvature tensor:
# R_{abcd} = \frac{1}{2} (\gamma_{ad,cb}+\gamma_{bc,da}-\gamma_{ac,bd}-\gamma_{bd,ac})
# + \gamma_{je} \Gamma^{j}_{bc}\Gamma^{e}_{ad} - \gamma_{je} \Gamma^{j}_{bd} \Gamma^{e}_{ac}
gammaDD_dDD = ixp.declarerank4("gammaDD_dDD", "sym01_sym23")
RiemannDDDD = ixp.zerorank4()
for a in range(DIM):
for b in range(DIM):
for c in range(DIM):
for d in range(DIM):
RiemannDDDD[a][b][c][d] = (gammaDD_dDD[a][d][c][b] + \
gammaDD_dDD[b][c][d][a] - \
gammaDD_dDD[a][c][b][d] - \
gammaDD_dDD[b][d][a][c]) / 2
for e in range(DIM):
for j in range(DIM):
RiemannDDDD[a][b][c][d] += gammaDD[j][e] * GammaUDD[j][b][c] * GammaUDD[e][a][d] - \
gammaDD[j][e] * GammaUDD[j][b][d] * GammaUDD[e][a][c]
# Step 6.b: We also need the extrinsic curvature tensor $K_{ij}$.
# In Cartesian coordinates, we already made the components gridfunctions.
# We will, however, need to calculate the trace of K seperately:
trK = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
trK += gammaUU[i][j] * kDD[i][j]
# Step 7: Build the formula for \psi_4.
# Gauss equation: involving the Riemann tensor and extrinsic curvature.
# GaussDDDD[i][j][k][l] =& R_{ijkl} + 2K_{i[k}K_{l]j}
GaussDDDD = ixp.zerorank4()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
GaussDDDD[i][j][k][l] = RiemannDDDD[i][j][k][
l] + kDD[i][k] * kDD[l][j] - kDD[i][l] * kDD[k][j]
# Codazzi equation: involving partial derivatives of the extrinsic curvature.
# We will first need to declare derivatives of kDD
# CodazziDDD[j][k][l] =& -2 (K_{j[k,l]} + \Gamma^p_{j[k} K_{l]p})
kDD_dD = ixp.declarerank3("kDD_dD", "sym01")
CodazziDDD = ixp.zerorank3()
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
CodazziDDD[j][k][l] = kDD_dD[j][l][k] - kDD_dD[j][k][l]
for p in range(DIM):
CodazziDDD[j][k][l] += GammaUDD[p][j][l] * kDD[k][
p] - GammaUDD[p][j][k] * kDD[l][p]
# Another piece. While not associated with any particular equation,
# this is still useful for organizational purposes.
# RojoDD[j][l]} = & R_{jl} - K_{jp} K^p_l + KK_{jl} \\
# = & \gamma^{pd} R_{jpld} - K_{jp} K^p_l + KK_{jl}
RojoDD = ixp.zerorank2()
for j in range(DIM):
for l in range(DIM):
RojoDD[j][l] = trK * kDD[j][l]
for p in range(DIM):
for d in range(DIM):
RojoDD[j][l] += gammaUU[p][d] * RiemannDDDD[j][p][l][
d] - kDD[j][p] * gammaUU[p][d] * kDD[d][l]
# Now we can calculate $\psi_4$ itself! We assume l^0 = n^0 = \frac{1}{\sqrt{2}}
# and m^0 = \overset{*}{m}{}^0 = 0 to simplify these equations.
# We calculate the Weyl scalars as defined in https://arxiv.org/abs/gr-qc/0104063
# In terms of the above-defined quantites, the psis are defined as:
# \psi_4 =&\ (\text{GaussDDDD[i][j][k][l]}) n^i \overset{*}{m}{}^j n^k \overset{*}{m}{}^l \\
# &+2 (\text{CodazziDDD[j][k][l]}) n^{0} \overset{*}{m}{}^{j} n^k \overset{*}{m}{}^l \\
# &+ (\text{RojoDD[j][l]}) n^{0} \overset{*}{m}{}^{j} n^{0} \overset{*}{m}{}^{l}.
# \psi_3 =&\ (\text{GaussDDDD[i][j][k][l]}) l^i n^j \overset{*}{m}{}^k n^l \\
# &+ (\text{CodazziDDD[j][k][l]}) (l^{0} n^{j} \overset{*}{m}{}^k n^l - l^{j} n^{0} \overset{*}{m}{}^k n^l - l^k n^j\overset{*}{m}{}^l n^0) \\
# &- (\text{RojoDD[j][l]}) l^{0} n^{j} \overset{*}{m}{}^l n^0 - l^{j} n^{0} \overset{*}{m}{}^l n^0 \\
# \psi_2 =&\ (\text{GaussDDDD[i][j][k][l]}) l^i m^j \overset{*}{m}{}^k n^l \\
# &+ (\text{CodazziDDD[j][k][l]}) (l^{0} m^{j} \overset{*}{m}{}^k n^l - l^{j} m^{0} \overset{*}{m}{}^k n^l - l^k m^l \overset{*}{m}{}^l n^0) \\
# &- (\text{RojoDD[j][l]}) l^0 m^j \overset{*}{m}{}^l n^0 \\
# \psi_1 =&\ (\text{GaussDDDD[i][j][k][l]}) n^i l^j m^k l^l \\
# &+ (\text{CodazziDDD[j][k][l]}) (n^{0} l^{j} m^k l^l - n^{j} l^{0} m^k l^l - n^k l^l m^j l^0) \\
# &- (\text{RojoDD[j][l]}) (n^{0} l^{j} m^l l^0 - n^{j} l^{0} m^l l^0) \\
# \psi_0 =&\ (\text{GaussDDDD[i][j][k][l]}) l^i m^j l^k m^l \\
# &+2 (\text{CodazziDDD[j][k][l]}) (l^0 m^j l^k m^l + l^k m^l l^0 m^j) \\
# &+ (\text{RojoDD[j][l]}) l^0 m^j l^0 m^j. \\
psi4r = sp.sympify(0)
psi4i = sp.sympify(0)
psi3r = sp.sympify(0)
psi3i = sp.sympify(0)
psi2r = sp.sympify(0)
psi2i = sp.sympify(0)
psi1r = sp.sympify(0)
psi1i = sp.sympify(0)
psi0r = sp.sympify(0)
psi0i = sp.sympify(0)
for l in range(DIM):
for j in range(DIM):
psi4r += RojoDD[j][l] * nn * nn * (remtetU[j] * remtetU[l] -
immtetU[j] * immtetU[l])
psi4i += RojoDD[j][l] * nn * nn * (-remtetU[j] * immtetU[l] -
immtetU[j] * remtetU[l])
psi3r += -RojoDD[j][l] * nn * nn * (ntetU[j] -
ltetU[j]) * remtetU[l]
psi3i += RojoDD[j][l] * nn * nn * (ntetU[j] -
ltetU[j]) * immtetU[l]
psi2r += -RojoDD[j][l] * nn * nn * (remtetU[l] * remtetU[j] +
immtetU[j] * immtetU[l])
psi2i += -RojoDD[j][l] * nn * nn * (immtetU[l] * remtetU[j] -
remtetU[j] * immtetU[l])
psi1r += RojoDD[j][l] * nn * nn * (ntetU[j] * remtetU[l] -
ltetU[j] * remtetU[l])
psi1i += RojoDD[j][l] * nn * nn * (ntetU[j] * immtetU[l] -
ltetU[j] * immtetU[l])
psi0r += RojoDD[j][l] * nn * nn * (remtetU[j] * remtetU[l] -
immtetU[j] * immtetU[l])
psi0i += RojoDD[j][l] * nn * nn * (remtetU[j] * immtetU[l] +
immtetU[j] * remtetU[l])
for l in range(DIM):
for j in range(DIM):
for k in range(DIM):
psi4r += 2 * CodazziDDD[j][k][l] * ntetU[k] * nn * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi4i += 2 * CodazziDDD[j][k][l] * ntetU[k] * nn * (
-remtetU[j] * immtetU[l] - immtetU[j] * remtetU[l])
psi3r += 1 * CodazziDDD[j][k][l] * nn * (
(ntetU[j] - ltetU[j]) * remtetU[k] * ntetU[l] -
remtetU[j] * ltetU[k] * ntetU[l])
psi3i += -1 * CodazziDDD[j][k][l] * nn * (
(ntetU[j] - ltetU[j]) * immtetU[k] * ntetU[l] -
immtetU[j] * ltetU[k] * ntetU[l])
psi2r += 1 * CodazziDDD[j][k][l] * nn * (
ntetU[l] *
(remtetU[j] * remtetU[k] + immtetU[j] * immtetU[k]) -
ltetU[k] *
(remtetU[j] * remtetU[l] + immtetU[j] * immtetU[l]))
psi2i += 1 * CodazziDDD[j][k][l] * nn * (
ntetU[l] *
(immtetU[j] * remtetU[k] - remtetU[j] * immtetU[k]) -
ltetU[k] *
(remtetU[j] * immtetU[l] - immtetU[j] * remtetU[l]))
psi1r += 1 * CodazziDDD[j][k][l] * nn * (
ltetU[j] * remtetU[k] * ltetU[l] - remtetU[j] * ntetU[k] *
ltetU[l] - ntetU[j] * remtetU[k] * ltetU[l])
psi1i += 1 * CodazziDDD[j][k][l] * nn * (
ltetU[j] * immtetU[k] * ltetU[l] - immtetU[j] * ntetU[k] *
ltetU[l] - ntetU[j] * immtetU[k] * ltetU[l])
psi0r += 2 * CodazziDDD[j][k][l] * nn * ltetU[k] * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi0i += 2 * CodazziDDD[j][k][l] * nn * ltetU[k] * (
remtetU[j] * immtetU[l] + immtetU[j] * remtetU[l])
for l in range(DIM):
for j in range(DIM):
for k in range(DIM):
for i in range(DIM):
psi4r += GaussDDDD[i][j][k][l] * ntetU[i] * ntetU[k] * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi4i += GaussDDDD[i][j][k][l] * ntetU[i] * ntetU[k] * (
-remtetU[j] * immtetU[l] - immtetU[j] * remtetU[l])
psi3r += GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[
j] * remtetU[k] * ntetU[l]
psi3i += -GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[
j] * immtetU[k] * ntetU[l]
psi2r += GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[l] * (
remtetU[j] * remtetU[k] + immtetU[j] * immtetU[k])
psi2i += GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[l] * (
immtetU[j] * remtetU[k] - remtetU[j] * immtetU[k])
psi1r += GaussDDDD[i][j][k][l] * ntetU[i] * ltetU[
j] * remtetU[k] * ltetU[l]
psi1i += GaussDDDD[i][j][k][l] * ntetU[i] * ltetU[
j] * immtetU[k] * ltetU[l]
psi0r += GaussDDDD[i][j][k][l] * ltetU[i] * ltetU[k] * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi0i += GaussDDDD[i][j][k][l] * ltetU[i] * ltetU[k] * (
remtetU[j] * immtetU[l] + immtetU[j] * remtetU[l])
def test_straight_line():
F = f(x)
Fd = F.diff(x)
L = sqrt(1 + Fd**2)
assert diff(L, F) == 0
assert diff(L, Fd) == Fd / sqrt(1 + Fd**2)
def test_issue_14112():
assert Sum((-1)**n/sqrt(n), (n, 1, oo)).is_absolutely_convergent() is S.false
assert Sum((-1)**(2*n)/n, (n, 1, oo)).is_convergent() is S.false
assert Sum((-2)**n + (-3)**n, (n, 1, oo)).is_convergent() is S.false
def test_bug1():
e = sqrt(-log(w))
assert e.subs(log(w), -x) == sqrt(x)
e = sqrt(-5 * log(w))
assert e.subs(log(w), -x) == sqrt(5 * x)
def test_hyperexpand_parametric():
assert hyperexpand(hyper([a, S(1)/2 + a], [S(1)/2], z)) \
== (1 + sqrt(z))**(-2*a)/2 + (1 - sqrt(z))**(-2*a)/2
assert hyperexpand(hyper([a, -S(1)/2 + a], [2*a], z)) \
== 2**(2*a - 1)*((-z + 1)**(S(1)/2) + 1)**(-2*a + 1)
def test_is_convergent():
# divergence tests --
assert Sum(n/(2*n + 1), (n, 1, oo)).is_convergent() is S.false
assert Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent() is S.false
assert Sum(3**(-2*n - 1)*n**n, (n, 1, oo)).is_convergent() is S.false
assert Sum((-1)**n*n, (n, 3, oo)).is_convergent() is S.false
assert Sum((-1)**n, (n, 1, oo)).is_convergent() is S.false
assert Sum(log(1/n), (n, 2, oo)).is_convergent() is S.false
# root test --
assert Sum((-12)**n/n, (n, 1, oo)).is_convergent() is S.false
# integral test --
# p-series test --
assert Sum(1/(n**2 + 1), (n, 1, oo)).is_convergent() is S.true
assert Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent() is S.true
assert Sum(2/(n*sqrt(n - 1)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(sqrt(n)*sqrt(n)), (n, 2, oo)).is_convergent() is S.false
# comparison test --
assert Sum(1/(n + log(n)), (n, 1, oo)).is_convergent() is S.false
assert Sum(1/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*log(n)), (n, 2, oo)).is_convergent() is S.false
assert Sum(2/(n*log(n)*log(log(n))**2), (n, 5, oo)).is_convergent() is S.true
assert Sum(2/(n*log(n)**2), (n, 2, oo)).is_convergent() is S.true
assert Sum((n - 1)/(n**2*log(n)**3), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*log(n)*log(log(n))), (n, 5, oo)).is_convergent() is S.false
assert Sum((n - 1)/(n*log(n)**3), (n, 3, oo)).is_convergent() is S.false
assert Sum(2/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*sqrt(log(n))*log(log(n))), (n, 100, oo)).is_convergent() is S.false
assert Sum(log(log(n))/(n*log(n)**2), (n, 100, oo)).is_convergent() is S.true
assert Sum(log(n)/n**2, (n, 5, oo)).is_convergent() is S.true
# alternating series tests --
assert Sum((-1)**(n - 1)/(n**2 - 1), (n, 3, oo)).is_convergent() is S.true
# with -negativeInfinite Limits
assert Sum(1/(n**2 + 1), (n, -oo, 1)).is_convergent() is S.true
assert Sum(1/(n - 1), (n, -oo, -1)).is_convergent() is S.false
assert Sum(1/(n**2 - 1), (n, -oo, -5)).is_convergent() is S.true
assert Sum(1/(n**2 - 1), (n, -oo, 2)).is_convergent() is S.true
assert Sum(1/(n**2 - 1), (n, -oo, oo)).is_convergent() is S.true
# piecewise functions
f = Piecewise((n**(-2), n <= 1), (n**2, n > 1))
assert Sum(f, (n, 1, oo)).is_convergent() is S.false
assert Sum(f, (n, -oo, oo)).is_convergent() is S.false
#assert Sum(f, (n, -oo, 1)).is_convergent() is S.true
# integral test
assert Sum(log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
assert Sum(-log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
# the following function has maxima located at (x, y) =
# (1.2, 0.43), (3.0, -0.25) and (6.8, 0.050)
eq = (x - 2)*(x**2 - 6*x + 4)*exp(-x)
assert Sum(eq, (x, 1, oo)).is_convergent() is S.true
assert Sum(eq, (x, 1, 2)).is_convergent() is S.true
assert Sum(1/(x**3), (x, 1, oo)).is_convergent() is S.true
assert Sum(1/(x**(S(1)/2)), (x, 1, oo)).is_convergent() is S.false
def normal(x, mu, sigma):
return 1 / sqrt(2 * pi * sigma**2) * exp(-(x - mu)**2 / 2 / sigma**2)
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z)* \
lowergamma(a - 1, z) - 1
# TODO [a+1, a-S.Half], [2*a]
assert hyperexpand(hyper([1, 2], [3],
z)) == -2 / z - 2 * log(-z + 1) / z**2
assert hyperexpand(hyper([S.Half, 2], [S(3)/2], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([S(1)/2, S(1)/2], [S(5)/2], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**(S(3)/2))
assert hyperexpand(hyper([1, 2], [S(3)/2], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([-S.Half - 1, 1, 2], [S.Half, 3], z)) == \
sqrt(z)*(6*z/7 - S(6)/5)*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + S(1)/2)/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [S.Half, a], [S(3)/2, a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + S(1)/2)*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_meijerint():
from sympy import symbols, expand, arg
s, t, mu = symbols('s t mu', real=True)
assert integrate(
meijerg([], [], [0], [], s * t) *
meijerg([], [], [mu / 2], [-mu / 2], t**2 / 4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
gamma(s + 1)
assert integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(
integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=False), Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1)**3 * exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x - mu) / (2 * sigma))**2), x, 0, oo)
assert simplify(i) == sqrt(pi) * sigma * (erf(mu / (2 * sigma)) + 1)
assert c == True
i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1 / (mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
(sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(
exp(-((x - mu) / sigma)**2 / 2) / sqrt(2 * pi * sigma**2), x, -oo,
oo) == (1, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x) * sin(x), x, 0, oo) == (S(1) / 2, True)
# Test a bug
def res(n):
return (1 / (1 + x**2)).diff(x, n).subs(x, 1) * (-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x) * sin(x + a), (x, 0, oo),
meijerg=True)) == sqrt(2) * sin(a + pi / 4) / 2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
from sympy import And, re
assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4)
*meijerg([], [], [b/2], [-b/2], x/4)*x**(s - 1), x, 0, oo) == \
(4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/(gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
*gamma(a/2 + b/2 - s + 1)),
And(0 < -2*re(4*s) + 8, 0 < re(a/2 + b/2 + s), re(2*s) < 1))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, S(1)/2)/4).expand()
# Test hyperexpand bug.
from sympy import lowergamma
n = symbols('n', integer=True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + S(1)/2,
alpha/2 + 1)), ((0, 0, S(1)/2), (-S(1)/2,)), alpha**S(2)/16)/4, True)
# test a bug related to 3016
a, s = symbols('a s', positive=True)
assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
a**(-s/2 - S(1)/2)*((-1)**s + 1)*gamma(s/2 + S(1)/2)/2
def test_issue_6122():
assert integrate(exp(-I*x**2), (x, -oo, oo), meijerg=True) == \
-I*sqrt(pi)*exp(I*pi/4)
def test_slow_general_univariate():
r = rootof(x**5 - x**2 + 1, 0)
assert solve(sqrt(x) + 1/root(x, 3) > 1) == \
Or(And(S(0) < x, x < r**6), And(r**6 < x, x < oo))
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y
from sympy import symbols, Symbol, Abs, expand_mul, combsimp, powsimp, sin
mu1, mu2 = symbols('mu1 mu2', real=True, nonzero=True, finite=True)
sigma1, sigma2 = symbols('sigma1 sigma2',
real=True,
nonzero=True,
finite=True,
positive=True)
rate = Symbol('lambda', real=True, positive=True, finite=True)
def normal(x, mu, sigma):
return 1 / sqrt(2 * pi * sigma**2) * exp(-(x - mu)**2 / 2 / sigma**2)
def exponential(x, rate):
return rate * exp(-rate * x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == \
mu1
assert integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**2 + sigma1**2
assert integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**3 + 3*mu1*sigma1**2
assert integrate(normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == 1
assert integrate(x * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu1
assert integrate(y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu2
assert integrate(x * y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu1 * mu2
assert integrate(
(x + y + 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == 1 + mu1 + mu2
assert integrate((x + y - 1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
-1 + mu1 + mu2
i = integrate(x**2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1**2 + sigma1**2
assert integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
sigma2**2 + mu2**2
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True) == \
1/rate
assert integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True) == \
2/rate**2
def E(expr):
res1 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo),
meijerg=True)
res2 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo),
meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x * y) == mu1 / rate
assert E(x * y**2) == mu1**2 / rate + sigma1**2 / rate
ans = sigma1**2 + 1 / rate**2
assert simplify(E((x + y + 1)**2) - E(x + y + 1)**2) == ans
assert simplify(E((x + y - 1)**2) - E(x + y - 1)**2) == ans
assert simplify(E((x + y)**2) - E(x + y)**2) == ans
# Beta' distribution
alpha, beta = symbols('alpha beta', positive=True)
betadist = x**(alpha - 1)*(1 + x)**(-alpha - beta)*gamma(alpha + beta) \
/gamma(alpha)/gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x * betadist, (x, 0, oo), meijerg=True, conds='separate')
assert (combsimp(i[0]), i[1]) == (alpha / (beta - 1), 1 < beta)
j = integrate(x**2 * betadist, (x, 0, oo), meijerg=True, conds='separate')
assert j[1] == (1 < beta - 1)
assert combsimp(j[0] - i[0]**2) == (alpha + beta - 1)*alpha \
/(beta - 2)/(beta - 1)**2
# Beta distribution
# NOTE: this is evaluated using antiderivatives. It also tests that
# meijerint_indefinite returns the simplest possible answer.
a, b = symbols('a b', positive=True)
betadist = x**(a - 1) * (-x + 1)**(b - 1) * gamma(a + b) / (gamma(a) *
gamma(b))
assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
assert simplify(integrate(x*betadist, (x, 0, 1), meijerg=True)) == \
a/(a + b)
assert simplify(integrate(x**2*betadist, (x, 0, 1), meijerg=True)) == \
a*(a + 1)/(a + b)/(a + b + 1)
assert simplify(integrate(x**y*betadist, (x, 0, 1), meijerg=True)) == \
gamma(a + b)*gamma(a + y)/gamma(a)/gamma(a + b + y)
# Chi distribution
k = Symbol('k', integer=True, positive=True)
chi = 2**(1 - k / 2) * x**(k - 1) * exp(-x**2 / 2) / gamma(k / 2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chi, (x, 0, oo), meijerg=True)) == \
sqrt(2)*gamma((k + 1)/2)/gamma(k/2)
assert simplify(integrate(x**2 * chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2**(-k / 2) / gamma(k / 2) * x**(k / 2 - 1) * exp(-x / 2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x**2*chisquared, (x, 0, oo), meijerg=True)) == \
k*(k + 2)
assert combsimp(
integrate(((x - k) / sqrt(2 * k))**3 * chisquared, (x, 0, oo),
meijerg=True)) == 2 * sqrt(2) / sqrt(k)
# Dagum distribution
a, b, p = symbols('a b p', positive=True)
# XXX (x/b)**a does not work
dagum = a * p / x * (x / b)**(a * p) / (1 + x**a / b**a)**(p + 1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
arg = x * dagum
assert simplify(integrate(
arg, (x, 0, oo), meijerg=True,
conds='none')) == a * b * gamma(1 - 1 / a) * gamma(p + 1 + 1 / a) / (
(a * p + 1) * gamma(p))
assert simplify(integrate(
x * arg, (x, 0, oo), meijerg=True,
conds='none')) == a * b**2 * gamma(1 -
2 / a) * gamma(p + 1 + 2 / a) / (
(a * p + 2) * gamma(p))
# F-distribution
d1, d2 = symbols('d1 d2', positive=True)
f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1 + d2))/x \
/gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x * f, (x, 0, oo), meijerg=True,
conds='none')) == d2 / (d2 - 2)
assert simplify(
integrate(x**2 * f, (x, 0, oo), meijerg=True,
conds='none')) == d2**2 * (d1 + 2) / d1 / (d2 - 4) / (d2 - 2)
# TODO gamma, rayleigh
# inverse gaussian
lamda, mu = symbols('lamda mu', positive=True)
dist = sqrt(lamda / 2 / pi) * x**(-S(3) / 2) * exp(
-lamda * (x - mu)**2 / x / 2 / mu**2)
mysimp = lambda expr: simplify(expr.rewrite(exp))
assert mysimp(integrate(dist, (x, 0, oo))) == 1
assert mysimp(integrate(x * dist, (x, 0, oo))) == mu
assert mysimp(integrate((x - mu)**2 * dist, (x, 0, oo))) == mu**3 / lamda
assert mysimp(integrate((x - mu)**3 * dist,
(x, 0, oo))) == 3 * mu**5 / lamda**2
# Levi
c = Symbol('c', positive=True)
assert integrate(
sqrt(c / 2 / pi) * exp(-c / 2 / (x - mu)) / (x - mu)**S('3/2'),
(x, mu, oo)) == 1
# higher moments oo
# log-logistic
distn = (beta/alpha)*x**(beta - 1)/alpha**(beta - 1)/ \
(1 + x**beta/alpha**beta)**2
assert simplify(integrate(distn, (x, 0, oo))) == 1
# NOTE the conditions are a mess, but correctly state beta > 1
assert simplify(integrate(x*distn, (x, 0, oo), conds='none')) == \
pi*alpha/beta/sin(pi/beta)
# (similar comment for conditions applies)
assert simplify(integrate(x**y*distn, (x, 0, oo), conds='none')) == \
pi*alpha**y*y/beta/sin(pi*y/beta)
# weibull
k = Symbol('k', positive=True)
n = Symbol('n', positive=True)
distn = k / lamda * (x / lamda)**(k - 1) * exp(-(x / lamda)**k)
assert simplify(integrate(distn, (x, 0, oo))) == 1
assert simplify(integrate(x**n*distn, (x, 0, oo))) == \
lamda**n*gamma(1 + n/k)
# rice distribution
from sympy import besseli
nu, sigma = symbols('nu sigma', positive=True)
rice = x / sigma**2 * exp(-(x**2 + nu**2) / 2 / sigma**2) * besseli(
0, x * nu / sigma**2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol('mu', real=True)
b = Symbol('b', positive=True)
laplace = exp(-abs(x - mu) / b) / 2 / b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x * laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x**2*laplace, (x, -oo, oo), meijerg=True) == \
2*b**2 + mu**2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol('k', positive=True)
assert combsimp(
expand_mul(
integrate(log(x) * x**(k - 1) * exp(-x) / gamma(k),
(x, 0, oo)))) == polygamma(0, k)
def test_issue_6343():
eq = -3 * x**2 / 2 - 45 * x / 4 + S(33) / 2 > 0
assert reduce_inequalities(eq) == \
And(x < -S(15)/4 + sqrt(401)/4, -sqrt(401)/4 - S(15)/4 < x)
def test_issue_8545():
eq = 1 - x - abs(1 - x)
ans = And(Lt(1, x), Lt(x, oo))
assert reduce_abs_inequality(eq, '<', x) == ans
eq = 1 - x - sqrt((1 - x)**2)
assert reduce_inequalities(eq < 0) == ans
def get_pure_l2_norm(alpha, l):
"""Compute the norm of a pure gaussian primitive."""
return sp.sqrt(
int(fac2(2 * l - 1)) / (2 * alpha / sp.pi)**sp.Rational(3, 2) /
(4 * alpha)**l)
def test_reduce_poly_inequalities_real_interval():
assert reduce_rational_inequalities([[Eq(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_rational_inequalities([[Le(x**2, 0)]], x,
relational=False) == FiniteSet(0)
assert reduce_rational_inequalities([[Lt(x**2, 0)]], x,
relational=False) == S.EmptySet
assert reduce_rational_inequalities(
[[Ge(x**2, 0)]], x, relational=False) == \
S.Reals if x.is_real else Interval(-oo, oo)
assert reduce_rational_inequalities(
[[Gt(x**2, 0)]], x, relational=False) == \
FiniteSet(0).complement(S.Reals)
assert reduce_rational_inequalities(
[[Ne(x**2, 0)]], x, relational=False) == \
FiniteSet(0).complement(S.Reals)
assert reduce_rational_inequalities([[Eq(x**2, 1)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_rational_inequalities([[Le(x**2, 1)]], x,
relational=False) == Interval(-1, 1)
assert reduce_rational_inequalities([[Lt(x**2, 1)]], x,
relational=False) == Interval(
-1, 1, True, True)
assert reduce_rational_inequalities(
[[Ge(x**2, 1)]], x, relational=False) == \
Union(Interval(-oo, -1), Interval(1, oo))
assert reduce_rational_inequalities(
[[Gt(x**2, 1)]], x, relational=False) == \
Interval(-1, 1).complement(S.Reals)
assert reduce_rational_inequalities(
[[Ne(x**2, 1)]], x, relational=False) == \
FiniteSet(-1, 1).complement(S.Reals)
assert reduce_rational_inequalities([[Eq(x**2, 1.0)]], x,
relational=False) == FiniteSet(
-1.0, 1.0).evalf()
assert reduce_rational_inequalities([[Le(x**2, 1.0)]], x,
relational=False) == Interval(
-1.0, 1.0)
assert reduce_rational_inequalities([[Lt(x**2, 1.0)]], x,
relational=False) == Interval(
-1.0, 1.0, True, True)
assert reduce_rational_inequalities(
[[Ge(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0), Interval(1.0, inf))
assert reduce_rational_inequalities(
[[Gt(x**2, 1.0)]], x, relational=False) == \
Union(Interval(-inf, -1.0, right_open=True),
Interval(1.0, inf, left_open=True))
assert reduce_rational_inequalities([[Ne(
x**2, 1.0)]], x, relational=False) == \
FiniteSet(-1.0, 1.0).complement(S.Reals)
s = sqrt(2)
assert reduce_rational_inequalities(
[[Lt(x**2 - 1, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == S.EmptySet
assert reduce_rational_inequalities(
[[Le(x**2 - 1, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == FiniteSet(-1, 1)
assert reduce_rational_inequalities(
[[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, False, False),
Interval(1, s, False, False))
assert reduce_rational_inequalities(
[[Le(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, False, True),
Interval(1, s, True, False))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, False),
Interval(1, s, False, True))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, True),
Interval(1, s, True, True))
assert reduce_rational_inequalities(
[[Lt(x**2 - 2, 0), Ne(x**2 - 1, 0)]], x,
relational=False) == Union(Interval(-s, -1, True, True),
Interval(-1, 1, True, True),
Interval(1, s, True, True))
],
[
diff(theta2_dot, x1),
diff(theta2_dot, y1),
diff(theta2_dot, theta1),
diff(theta2_dot, x2),
diff(theta2_dot, y2),
diff(theta2_dot, theta2)
]])
G = Matrix([[cos(theta1), 0, 0, 0], [sin(theta2), 0, 0,
0], [0, 1, 0, 0],
[0, 0, cos(theta2), 0], [0, 0, sin(theta2), 0],
[0, 0, 0, 1]])
h = sqrt((x1 - x2)**2 + (y1 - y2)**2)
H = Matrix([[
diff(h, x1),
diff(h, y1),
diff(h, theta1),
diff(h, x2),
diff(h, y2),
diff(h, theta2)
]])
Q = Dt * diag(sigv**2, sigw**2, sigv**2, sigw**2)
R = Dt * diag(sigr**2)
P_dot = (F * P + P * F.T - P * H.T * (R**-1) * H * P + G * Q * G.T)
def test_reduce_inequalities_general():
assert reduce_inequalities(Ge(sqrt(2) * x,
1)) == And(sqrt(2) / 2 <= x, x < oo)
assert reduce_inequalities(PurePoly(x + 1, x) > 0) == And(
S(-1) < x, x < oo)
def main():
x = Symbol("x")
# mplot2d(log(x), (x, 0, 2, 100))
# mplot2d([sin(x), -sin(x)], (x, float(-2*pi), float(2*pi), 50))
mplot2d([sqrt(x), -sqrt(x), sqrt(-x), -sqrt(-x)], (x, -40.0, 40.0, 80))
def get_cart_l2_norm(alpha, nx, ny, nz):
"""Compute the norm of a Cartesian gaussian primitive."""
return sp.sqrt(
int(fac2(2 * nx - 1) * fac2(2 * ny - 1) * fac2(2 * nz - 1)) /
(2 * alpha / sp.pi)**sp.Rational(3, 2) / (4 * alpha)**(nx + ny + nz))
