def test_sympy_parser():
x = Symbol('x')
inputs = {
'2*x': 2 * x,
'3.00': Float(3),
'22/7': Rational(22, 7),
'2+3j': 2 + 3*I,
'exp(x)': exp(x),
'x!': factorial(x),
'x!!': factorial2(x),
'(x + 1)! - 1': factorial(x + 1) - 1,
'3.[3]': Rational(10, 3),
'.0[3]': Rational(1, 30),
'3.2[3]': Rational(97, 30),
'1.3[12]': Rational(433, 330),
'1 + 3.[3]': Rational(13, 3),
'1 + .0[3]': Rational(31, 30),
'1 + 3.2[3]': Rational(127, 30),
'.[0011]': Rational(1, 909),
'0.1[00102] + 1': Rational(366697, 333330),
'1.[0191]': Rational(10190, 9999),
'10!': 3628800,
'-(2)': -Integer(2),
'[-1, -2, 3]': [Integer(-1), Integer(-2), Integer(3)],
'Symbol("x").free_symbols': x.free_symbols,
"S('S(3).n(n=3)')": 3.00,
'factorint(12, visual=True)': Mul(
Pow(2, 2, evaluate=False),
Pow(3, 1, evaluate=False),
evaluate=False),
'Limit(sin(x), x, 0, dir="-")': Limit(sin(x), x, 0, dir='-'),
}
for text, result in inputs.items():
assert parse_expr(text) == result
def test_Function():
assert mcode(f(x, y, z)) == "f[x, y, z]"
assert mcode(sin(x)**cos(x)) == "Sin[x]^Cos[x]"
assert mcode(conjugate(x)) == "Conjugate[x]"
assert mcode(Max(x, y, z) * Min(y, z)) == "Max[x, y, z]*Min[y, z]"
assert mcode(fresnelc(x)) == "FresnelC[x]"
assert mcode(fresnels(x)) == "FresnelS[x]"
assert mcode(gamma(x)) == "Gamma[x]"
assert mcode(uppergamma(x, y)) == "Gamma[x, y]"
assert mcode(polygamma(x, y)) == "PolyGamma[x, y]"
assert mcode(loggamma(x)) == "LogGamma[x]"
assert mcode(erf(x)) == "Erf[x]"
assert mcode(erfc(x)) == "Erfc[x]"
assert mcode(erfi(x)) == "Erfi[x]"
assert mcode(erf2(x, y)) == "Erf[x, y]"
assert mcode(expint(x, y)) == "ExpIntegralE[x, y]"
assert mcode(erfcinv(x)) == "InverseErfc[x]"
assert mcode(erfinv(x)) == "InverseErf[x]"
assert mcode(erf2inv(x, y)) == "InverseErf[x, y]"
assert mcode(Ei(x)) == "ExpIntegralEi[x]"
assert mcode(Ci(x)) == "CosIntegral[x]"
assert mcode(li(x)) == "LogIntegral[x]"
assert mcode(Si(x)) == "SinIntegral[x]"
assert mcode(Shi(x)) == "SinhIntegral[x]"
assert mcode(Chi(x)) == "CoshIntegral[x]"
assert mcode(beta(x, y)) == "Beta[x, y]"
assert mcode(factorial(x)) == "Factorial[x]"
assert mcode(factorial2(x)) == "Factorial2[x]"
assert mcode(subfactorial(x)) == "Subfactorial[x]"
assert mcode(FallingFactorial(x, y)) == "FactorialPower[x, y]"
assert mcode(RisingFactorial(x, y)) == "Pochhammer[x, y]"
assert mcode(catalan(x)) == "CatalanNumber[x]"
assert mcode(harmonic(x)) == "HarmonicNumber[x]"
assert mcode(harmonic(x, y)) == "HarmonicNumber[x, y]"
def R_nl(n, l, nu, r):
"""
Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic
oscillator.
Parameters
==========
``n`` :
The "nodal" quantum number. Corresponds to the number of nodes in
the wavefunction. ``n >= 0``
``l`` :
The quantum number for orbital angular momentum.
``nu`` :
mass-scaled frequency: nu = m*omega/(2*hbar) where `m` is the mass
and `omega` the frequency of the oscillator.
(in atomic units ``nu == omega/2``)
``r`` :
Radial coordinate.
Examples
========
>>> from sympy.physics.sho import R_nl
>>> from sympy.abc import r, nu, l
>>> R_nl(0, 0, 1, r)
2*2**(3/4)*exp(-r**2)/pi**(1/4)
>>> R_nl(1, 0, 1, r)
4*2**(1/4)*sqrt(3)*(3/2 - 2*r**2)*exp(-r**2)/(3*pi**(1/4))
l, nu and r may be symbolic:
>>> R_nl(0, 0, nu, r)
2*2**(3/4)*sqrt(nu**(3/2))*exp(-nu*r**2)/pi**(1/4)
>>> R_nl(0, l, 1, r)
r**l*sqrt(2**(l + 3/2)*2**(l + 2)/factorial2(2*l + 1))*exp(-r**2)/pi**(1/4)
The normalization of the radial wavefunction is:
>>> from sympy import Integral, oo
>>> Integral(R_nl(0, 0, 1, r)**2*r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 0, 1, r)**2*r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 1, 1, r)**2*r**2, (r, 0, oo)).n()
1.00000000000000
"""
n, l, nu, r = map(S, [n, l, nu, r])
# formula uses n >= 1 (instead of nodal n >= 0)
n = n + 1
C = sqrt(
((2*nu)**(l + Rational(3, 2))*2**(n + l + 1)*factorial(n - 1))/
(sqrt(pi)*(factorial2(2*n + 2*l - 1)))
)
return C*r**(l)*exp(-nu*r**2)*assoc_laguerre(n - 1, l + S.Half, 2*nu*r**2)
def R_nl(n, l, nu, r):
"""
Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic
oscillator.
``n``
the "nodal" quantum number. Corresponds to the number of nodes in
the wavefunction. n >= 0
``l``
the quantum number for orbital angular momentum
``nu``
mass-scaled frequency: nu = m*omega/(2*hbar) where `m` is the mass
and `omega` the frequency of the oscillator.
(in atomic units nu == omega/2)
``r``
Radial coordinate
Examples
========
>>> from sympy.physics.sho import R_nl
>>> from sympy import var
>>> var("r nu l")
(r, nu, l)
>>> R_nl(0, 0, 1, r)
2*2**(3/4)*exp(-r**2)/pi**(1/4)
>>> R_nl(1, 0, 1, r)
4*2**(1/4)*sqrt(3)*(-2*r**2 + 3/2)*exp(-r**2)/(3*pi**(1/4))
l, nu and r may be symbolic:
>>> R_nl(0, 0, nu, r)
2*2**(3/4)*sqrt(nu**(3/2))*exp(-nu*r**2)/pi**(1/4)
>>> R_nl(0, l, 1, r)
r**l*sqrt(2**(l + 3/2)*2**(l + 2)/factorial2(2*l + 1))*exp(-r**2)/pi**(1/4)
The normalization of the radial wavefunction is:
>>> from sympy import Integral, oo
>>> Integral(R_nl(0, 0, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 0, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 1, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
"""
n, l, nu, r = map(S, [n, l, nu, r])
# formula uses n >= 1 (instead of nodal n >= 0)
n = n + 1
C = sqrt(
((2 * nu) ** (l + Rational(3, 2)) * 2 ** (n + l + 1) * factorial(n - 1))
/ (sqrt(pi) * (factorial2(2 * n + 2 * l - 1)))
)
return C * r ** (l) * exp(-nu * r ** 2) * assoc_laguerre(n - 1, l + S(1) / 2, 2 * nu * r ** 2)
def R_nl(n, l, nu, r):
"""
Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic oscillator.
``n``
the "nodal" quantum number. Corresponds to the number of nodes in the
wavefunction. n >= 0
``l``
the quantum number for orbital angular momentum
``nu``
mass-scaled frequency: nu = m*omega/(2*hbar) where `m' is the mass and
`omega' the frequency of the oscillator. (in atomic units nu == omega/2)
``r``
Radial coordinate
:Examples:
>>> from sympy.physics.sho import R_nl
>>> from sympy import var
>>> var("r nu l")
(r, nu, l)
>>> R_nl(0, 0, 1, r)
2*2**(3/4)*exp(-r**2)/pi**(1/4)
>>> R_nl(1, 0, 1, r)
4*2**(1/4)*3**(1/2)*(3/2 - 2*r**2)*exp(-r**2)/(3*pi**(1/4))
l, nu and r may be symbolic:
>>> R_nl(0, 0, nu, r)
2*2**(3/4)*(nu**(3/2))**(1/2)*exp(-nu*r**2)/pi**(1/4)
>>> R_nl(0, l, 1, r)
r**l*(2**(2 + l)*2**(3/2 + l)/(1 + 2*l)!!)**(1/2)*exp(-r**2)/pi**(1/4)
The normalization of the radial wavefunction is::
>>> from sympy import Integral, oo
>>> Integral(R_nl(0, 0, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 0, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
>>> Integral(R_nl(1, 1, 1, r)**2 * r**2, (r, 0, oo)).n()
1.00000000000000
"""
n, l, nu, r = map(S, [n, l, nu, r])
# formula uses n >= 1 (instead of nodal n >= 0)
n = n + 1
C = sqrt(
((2 * nu)**(l + Rational(3, 2)) * 2**(n + l + 1) * factorial(n - 1)) /
(sqrt(pi) * (factorial2(2 * n + 2 * l - 1))))
return C * r**(l) * exp(-nu * r**2) * laguerre_l(n - 1, l + S(1) / 2,
2 * nu * r**2)
def test_sympy_parser():
x = Symbol("x")
inputs = {
"2*x":
2 * x,
"3.00":
Float(3),
"22/7":
Rational(22, 7),
"2+3j":
2 + 3 * I,
"exp(x)":
exp(x),
"x!":
factorial(x),
"x!!":
factorial2(x),
"(x + 1)! - 1":
factorial(x + 1) - 1,
"3.[3]":
Rational(10, 3),
".0[3]":
Rational(1, 30),
"3.2[3]":
Rational(97, 30),
"1.3[12]":
Rational(433, 330),
"1 + 3.[3]":
Rational(13, 3),
"1 + .0[3]":
Rational(31, 30),
"1 + 3.2[3]":
Rational(127, 30),
".[0011]":
Rational(1, 909),
"0.1[00102] + 1":
Rational(366697, 333330),
"1.[0191]":
Rational(10190, 9999),
"10!":
3628800,
"-(2)":
-Integer(2),
"[-1, -2, 3]": [Integer(-1), Integer(-2),
Integer(3)],
'Symbol("x").free_symbols':
x.free_symbols,
"S('S(3).n(n=3)')":
3.00,
"factorint(12, visual=True)":
Mul(Pow(2, 2, evaluate=False),
Pow(3, 1, evaluate=False),
evaluate=False),
'Limit(sin(x), x, 0, dir="-")':
Limit(sin(x), x, 0, dir="-"),
}
for text, result in inputs.items():
assert parse_expr(text) == result
raises(TypeError, lambda: parse_expr("x", standard_transformations))
raises(TypeError, lambda: parse_expr("x", transformations=lambda x, y: 1))
raises(TypeError,
lambda: parse_expr("x", transformations=(lambda x, y: 1,
)))
raises(TypeError, lambda: parse_expr("x", transformations=((), )))
raises(TypeError, lambda: parse_expr("x", {}, [], []))
raises(TypeError, lambda: parse_expr("x", [], [], {}))
raises(TypeError, lambda: parse_expr("x", [], [], {}))
