def test_inversion():
def inv(f):
return piecewise_fold(meijerint_inversion(f, s, t))
assert inv(1/(s**2 + 1)) == sin(t)*Heaviside(t)
assert inv(s/(s**2 + 1)) == cos(t)*Heaviside(t)
assert inv(exp(-s)/s) == Heaviside(t - 1)
assert inv(1/sqrt(1 + s**2)) == besselj(0, t)*Heaviside(t)
# Test some antcedents checking.
assert meijerint_inversion(sqrt(s)/sqrt(1 + s**2), s, t) is None
assert inv(exp(s**2)) is None
assert meijerint_inversion(exp(-s**2), s, t) is None
def test_fourier_transform():
from diofant import simplify, expand, expand_complex, factor, expand_trig
FT = fourier_transform
IFT = inverse_fourier_transform
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
def sinc(x):
return sin(pi * x) / (pi * x)
k = symbols('k', extended_real=True)
f = Function("f")
# TODO for this to work with real a, need to expand abs(a*x) to abs(a)*abs(x)
a = symbols('a', positive=True)
b = symbols('b', positive=True)
posk = symbols('posk', positive=True)
# Test unevaluated form
assert fourier_transform(f(x), x, k) == FourierTransform(f(x), x, k)
assert inverse_fourier_transform(f(k), k,
x) == InverseFourierTransform(f(k), k, x)
# basic examples from wikipedia
assert simp(FT(Heaviside(1 - abs(2 * a * x)), x, k)) == sinc(k / a) / a
# TODO IFT is a *mess*
assert simp(FT(Heaviside(1 - abs(a * x)) * (1 - abs(a * x)), x,
k)) == sinc(k / a)**2 / a
# TODO IFT
assert factor(FT(exp(-a*x)*Heaviside(x), x, k), extension=I) == \
1/(a + 2*pi*I*k)
# NOTE: the ift comes out in pieces
assert IFT(1 / (a + 2 * pi * I * x), x, posk,
noconds=False) == (exp(-a * posk), True)
assert IFT(1 / (a + 2 * pi * I * x), x, -posk, noconds=False) == (0, True)
assert IFT(1 / (a + 2 * pi * I * x),
x,
symbols('k', negative=True),
noconds=False) == (0, True)
# TODO IFT without factoring comes out as meijer g
assert factor(FT(x*exp(-a*x)*Heaviside(x), x, k), extension=I) == \
1/(a + 2*pi*I*k)**2
assert FT(exp(-a*x)*sin(b*x)*Heaviside(x), x, k) == \
b/(b**2 + (a + 2*I*pi*k)**2)
assert FT(exp(-a * x**2), x,
k) == sqrt(pi) * exp(-pi**2 * k**2 / a) / sqrt(a)
assert IFT(sqrt(pi / a) * exp(-(pi * k)**2 / a), k, x) == exp(-a * x**2)
assert FT(exp(-a * abs(x)), x, k) == 2 * a / (a**2 + 4 * pi**2 * k**2)
def test_Abs_rewrite():
x = Symbol('x', extended_real=True)
a = Abs(x).rewrite(Heaviside).expand()
assert a == x*Heaviside(x) - x*Heaviside(-x)
for i in [-2, -1, 0, 1, 2]:
assert a.subs(x, i) == abs(i)
y = Symbol('y')
assert Abs(y).rewrite(Heaviside) == Abs(y)
x, y = Symbol('x', extended_real=True), Symbol('y')
assert Abs(x).rewrite(Piecewise) == Piecewise((x, x >= 0), (-x, True))
assert Abs(y).rewrite(Piecewise) == Abs(y)
assert Abs(y).rewrite(sign) == y/sign(y)
def test_Max():
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
p = Symbol('p', positive=True)
r = Symbol('r', extended_real=True)
assert Max(5, 4) == 5
# lists
pytest.raises(ValueError, Max)
assert Max(x, y) == Max(y, x)
assert Max(x, y, z) == Max(z, y, x)
assert Max(x, Max(y, z)) == Max(z, y, x)
assert Max(x, Min(y, oo)) == Max(x, y)
assert Max(n, -oo, n_, p, 2) == Max(p, 2)
assert Max(n, -oo, n_, p) == p
assert Max(2, x, p, n, -oo, -oo, n_, p, 2) == Max(2, x, p)
assert Max(0, x, 1, y) == Max(1, x, y)
assert Max(r, r + 1, r - 1) == 1 + r
assert Max(1000, 100, -100, x, p, n) == Max(p, x, 1000)
assert Max(cos(x), sin(x)) == Max(sin(x), cos(x))
assert Max(cos(x), sin(x)).subs({x: 1}) == sin(1)
assert Max(cos(x), sin(x)).subs({x: Rational(1, 2)}) == cos(Rational(1, 2))
pytest.raises(ValueError, lambda: Max(cos(x), sin(x)).subs({x: I}))
pytest.raises(ValueError, lambda: Max(I))
pytest.raises(ValueError, lambda: Max(I, x))
pytest.raises(ValueError, lambda: Max(zoo, 1))
# interesting:
# Max(n, -oo, n_, p, 2) == Max(p, 2)
# True
# Max(n, -oo, n_, p, 1000) == Max(p, 1000)
# False
assert Max(1, x).diff(x) == Heaviside(x - 1)
assert Max(x, 1).diff(x) == Heaviside(x - 1)
assert Max(x**2, 1 + x, 1).diff(x) == \
2*x*Heaviside(x**2 - Max(1, x + 1)) \
+ Heaviside(x - Max(1, x**2) + 1)
pytest.raises(ArgumentIndexError, lambda: Max(1, x).fdiff(3))
a, b = Symbol('a', extended_real=True), Symbol('b', extended_real=True)
# a and b are both real, Max(a, b) should be real
assert Max(a, b).is_extended_real
# issue sympy/sympy#7233
e = Max(0, x)
assert e.evalf(strict=False).args == (0, x)
assert Max(2 * a * b * sqrt(2), a * b * 6).simplify() == 6 * a * b
def test_inverse_mellin_transform():
IMT = inverse_mellin_transform
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
(-x/2 + 1/(2*x))*Heaviside(-x + 1)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False),
force=True)).replace(exp_polar, exp)
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
* gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
def test_expint():
from diofant import E1, expint, Max, re, lerchphi, Symbol, simplify, Si, Ci, Ei
aneg = Symbol('a', negative=True)
u = Symbol('u', polar=True)
assert mellin_transform(E1(x), x, s) == (gamma(s) / s, (0, oo), True)
assert inverse_mellin_transform(gamma(s) / s, s, x,
(0, oo)).rewrite(expint).expand() == E1(x)
assert mellin_transform(expint(a, x), x, s) == \
(gamma(s)/(a + s - 1), (Max(1 - re(a), 0), oo), True)
# XXX IMT has hickups with complicated strips ...
assert simplify(unpolarify(
inverse_mellin_transform(gamma(s)/(aneg + s - 1), s, x,
(1 - aneg, oo)).rewrite(expint).expand(func=True))) == \
expint(aneg, x)
assert mellin_transform(Si(x), x, s) == \
(-2**s*sqrt(pi)*gamma(s/2 + Rational(1, 2))/(
2*s*gamma(-s/2 + 1)), (-1, 0), True)
assert inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)
/ (2*s*gamma(-s/2 + 1)), s, x, (-1, 0)) \
== Si(x)
assert mellin_transform(Ci(sqrt(x)), x, s) == \
(-2**(2*s - 1)*sqrt(pi)*gamma(s)/(s*gamma(-s + Rational(1, 2))), (0, 1), True)
assert inverse_mellin_transform(
-4**s * sqrt(pi) * gamma(s) / (2 * s * gamma(-s + Rational(1, 2))), s,
u, (0, 1)).expand() == Ci(sqrt(u))
# TODO LT of Si, Shi, Chi is a mess ...
assert laplace_transform(Ci(x), x, s) == (-log(1 + s**2) / 2 / s, 0, True)
assert laplace_transform(expint(a, x), x, s) == \
(lerchphi(s*polar_lift(-1), 1, a), 0, Integer(0) < re(a))
assert laplace_transform(expint(1, x), x, s) == (log(s + 1) / s, 0, True)
assert laplace_transform(expint(2, x), x, s) == \
((s - log(s + 1))/s**2, 0, True)
assert inverse_laplace_transform(-log(1 + s**2)/2/s, s, u).expand() == \
Heaviside(u)*Ci(u)
assert inverse_laplace_transform(log(s + 1)/s, s, x).rewrite(expint) == \
Heaviside(x)*E1(x)
assert inverse_laplace_transform((s - log(s + 1))/s**2, s,
x).rewrite(expint).expand() == \
(expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand()
def test_rewrite():
x = Symbol('x', extended_real=True)
assert Heaviside(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (Rational(1, 2), Eq(x, 0)), (0, True))
assert Heaviside(y).rewrite(Piecewise) == Heaviside(y)
assert Heaviside(x).rewrite(sign) == (sign(x)+1)/2
assert Heaviside(y).rewrite(sign) == Heaviside(y)
def test_inverse_mellin_transform():
IMT = inverse_mellin_transform
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
(-x/2 + 1/(2*x))*Heaviside(-x + 1)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False),
force=True)).replace(exp_polar, exp)
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
* gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
# issue sympy/sympy#8882
# This is the original test.
# r = Symbol('r')
# psi = 1/r*sin(r)*exp(-(a0*r))
# h = -1/2*diff(psi, r, r) - 1/r*psi
# f = 4*pi*psi*h*r**2
# assert integrate(f, (r, -oo, 3), meijerg=True).has(Integral) == True
# To save time, only the critical part is included.
F = -a**(-s + 1)*(4 + 1/a**2)**(-s/2)*sqrt(1/a**2)*exp(-s*I*pi) * \
sin(s*atan(sqrt(1/a**2)/2))*gamma(s)
pytest.raises(
IntegralTransformError, lambda: inverse_mellin_transform(
F, s, x, (-1, oo), **{
'as_meijerg': True,
'needeval': True
}))
def test_inverse_laplace_transform():
from diofant import sinh, cosh, besselj, besseli, simplify, factor_terms
ILT = inverse_laplace_transform
a, b, c, = symbols('a b c', positive=True, finite=True)
t = symbols('t')
def simp_hyp(expr):
return factor_terms(expand_mul(expr)).rewrite(sin)
# just test inverses of all of the above
assert ILT(1 / s, s, t) == Heaviside(t)
assert ILT(1 / s**2, s, t) == t * Heaviside(t)
assert ILT(1 / s**5, s, t) == t**4 * Heaviside(t) / 24
assert ILT(exp(-a * s) / s, s, t) == Heaviside(t - a)
assert ILT(exp(-a * s) / (s + b), s,
t) == exp(b * (a - t)) * Heaviside(-a + t)
assert ILT(a / (s**2 + a**2), s, t) == sin(a * t) * Heaviside(t)
assert ILT(s / (s**2 + a**2), s, t) == cos(a * t) * Heaviside(t)
# TODO is there a way around simp_hyp?
assert simp_hyp(ILT(a / (s**2 - a**2), s, t)) == sinh(a * t) * Heaviside(t)
assert simp_hyp(ILT(s / (s**2 - a**2), s, t)) == cosh(a * t) * Heaviside(t)
assert ILT(a / ((s + b)**2 + a**2), s,
t) == exp(-b * t) * sin(a * t) * Heaviside(t)
assert ILT((s + b) / ((s + b)**2 + a**2), s,
t) == exp(-b * t) * cos(a * t) * Heaviside(t)
# TODO sinh/cosh shifted come out a mess. also delayed trig is a mess
# TODO should this simplify further?
assert ILT(exp(-a*s)/s**b, s, t) == \
(t - a)**(b - 1)*Heaviside(t - a)/gamma(b)
assert ILT(exp(-a*s)/sqrt(1 + s**2), s, t) == \
Heaviside(t - a)*besselj(0, a - t) # note: besselj(0, x) is even
# XXX ILT turns these branch factor into trig functions ...
assert simplify(ILT(a**b*(s + sqrt(s**2 - a**2))**(-b)/sqrt(s**2 - a**2),
s, t).rewrite(exp)) == \
Heaviside(t)*besseli(b, a*t)
assert ILT(a**b*(s + sqrt(s**2 + a**2))**(-b)/sqrt(s**2 + a**2),
s, t).rewrite(exp) == \
Heaviside(t)*besselj(b, a*t)
assert ILT(1 / (s * sqrt(s + 1)), s, t) == Heaviside(t) * erf(sqrt(t))
# TODO can we make erf(t) work?
assert ILT(1 / (s**2 * (s**2 + 1)), s, t) == (t - sin(t)) * Heaviside(t)
assert ILT( (s * eye(2) - Matrix([[1, 0], [0, 2]])).inv(), s, t) ==\
Matrix([[exp(t)*Heaviside(t), 0], [0, exp(2*t)*Heaviside(t)]])
def test_laplace_transform():
from diofant import fresnels, fresnelc
LT = laplace_transform
a, b, c, = symbols('a b c', positive=True)
t = symbols('t')
w = Symbol("w")
f = Function("f")
# Test unevaluated form
assert laplace_transform(f(t), t, w) == LaplaceTransform(f(t), t, w)
assert inverse_laplace_transform(f(w), w, t,
plane=0) == InverseLaplaceTransform(
f(w), w, t, 0)
# test a bug
spos = symbols('s', positive=True)
assert LT(exp(t), t, spos)[:2] == (1 / (spos - 1), True)
# basic tests from wikipedia
assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \
((s + c)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True)
assert LT(t**a, t, s) == (s**(-a - 1) * gamma(a + 1), 0, True)
assert LT(Heaviside(t), t, s) == (1 / s, 0, True)
assert LT(Heaviside(t - a), t, s) == (exp(-a * s) / s, 0, True)
assert LT(1 - exp(-a * t), t, s) == (a / (s * (a + s)), 0, True)
assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \
== exp(-b)/(s**2 - 1)
assert LT(exp(t), t, s)[:2] == (1 / (s - 1), 1)
assert LT(exp(2 * t), t, s)[:2] == (1 / (s - 2), 2)
assert LT(exp(a * t), t, s)[:2] == (1 / (s - a), a)
assert LT(log(t / a), t,
s) == ((log(a * s) + EulerGamma) / s / -1, 0, True)
assert LT(erf(t), t, s) == ((-erf(s / 2) + 1) * exp(s**2 / 4) / s, 0, True)
assert LT(sin(a * t), t, s) == (a / (a**2 + s**2), 0, True)
assert LT(cos(a * t), t, s) == (s / (a**2 + s**2), 0, True)
# TODO would be nice to have these come out better
assert LT(exp(-a * t) * sin(b * t), t,
s) == (b / (b**2 + (a + s)**2), -a, True)
assert LT(exp(-a*t)*cos(b*t), t, s) == \
((a + s)/(b**2 + (a + s)**2), -a, True)
assert LT(besselj(0, t), t, s) == (1 / sqrt(1 + s**2), 0, True)
assert LT(besselj(1, t), t, s) == (1 - 1 / sqrt(1 + 1 / s**2), 0, True)
# TODO general order works, but is a *mess*
# TODO besseli also works, but is an even greater mess
# test a bug in conditions processing
# TODO the auxiliary condition should be recognised/simplified
assert LT(exp(t) * cos(t), t, s)[:-1] in [
((s - 1) / (s**2 - 2 * s + 2), -oo),
((s - 1) / ((s - 1)**2 + 1), -oo),
]
# Fresnel functions
assert laplace_transform(fresnels(t), t, s) == \
((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 -
cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True)
assert laplace_transform(
fresnelc(t), t,
s) == ((sin(s**2 / (2 * pi)) * fresnelc(s / pi) / s -
cos(s**2 / (2 * pi)) * fresnels(s / pi) / s +
sqrt(2) * cos(s**2 / (2 * pi) + pi / 4) / (2 * s), 0, True))
assert LT(Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]]), t, s) ==\
Matrix([
[(1/(s - 1), 1, True), ((s + 1)**(-2), 0, True)],
[((s + 1)**(-2), 0, True), (1/(s - 1), 1, True)]
])
def test_inverse_mellin_transform():
from diofant import (sin, simplify, Max, Min, expand, powsimp, exp_polar,
cos, cot)
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1 / x)
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
(x**2 + 1)*Heaviside(1 - x)/(4*x)
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
(-x/2 + 1/(2*x))*Heaviside(-x + 1)
# test expansion of sums
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1) * exp(-x) / x
# test factorisation of polys
r = symbols('r', extended_real=True)
assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
).subs(x, r).rewrite(sin).simplify() \
== sin(r)*Heaviside(1 - exp(-r))
# test multiplicative substitution
_a, _b = symbols('a b', positive=True)
assert IMT(_b**(-s / _a) * factorial(s / _a) / s, s, x,
(0, oo)) == exp(-_b * x**_a)
assert IMT(factorial(_a / _b + s / _b) / (_a + s), s, x,
(-_a, oo)) == x**_a * exp(-x**_b)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False),
force=True)).replace(exp_polar, exp)
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
nu = symbols('nu', extended_real=True, finite=True)
assert IMT(-1 / (nu + s), s, x, (-oo, None)) == x**nu * Heaviside(x - 1)
assert IMT(1 / (nu + s), s, x, (None, oo)) == x**nu * Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
== (1 - x)**(beta - 1)*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
s, x, (-oo, None))) \
== (x - 1)**(beta - 1)*Heaviside(x - 1)
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
== (1/(x + 1))**rho
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
* gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
* gamma(-c/2 - s)/gamma(1 - c - s),
s, x, (0, -re(c)/2))) == \
(1 + sqrt(x + 1))**c
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
/ gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
b**2 + x)/(b**2 + x)
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
b**c*(sqrt(1 + x/b**2) + 1)**c
# Section 8.4.5
assert IMT(24 / s**5, s, x, (0, oo)) == log(x)**4 * Heaviside(1 - x)
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
log(x)**3*Heaviside(x - 1)
assert IMT(pi / (s * sin(pi * s)), s, x, (-1, 0)) == log(x + 1)
assert IMT(pi / (s * sin(pi * s / 2)), s, x, (-2, 0)) == log(x**2 + 1)
assert IMT(pi / (s * sin(2 * pi * s)), s, x,
(-Rational(1, 2), 0)) == log(sqrt(x) + 1)
assert IMT(pi / (s * sin(pi * s)), s, x, (0, 1)) == log(1 + 1 / x)
# TODO
def mysimp(expr):
from diofant import expand, logcombine, powsimp
return expand(powsimp(logcombine(expr, force=True),
force=True,
deep=True),
force=True).replace(exp_polar, exp)
assert mysimp(mysimp(IMT(pi / (s * tan(pi * s)), s, x, (-1, 0)))) in [
log(1 - x) * Heaviside(1 - x) + log(x - 1) * Heaviside(x - 1),
log(x) * Heaviside(x - 1) + log(1 - 1 / x) * Heaviside(x - 1) +
log(-x + 1) * Heaviside(-x + 1)
]
# test passing cot
assert mysimp(IMT(pi * cot(pi * s) / s, s, x, (0, 1))) in [
log(1 / x - 1) * Heaviside(1 - x) + log(1 - 1 / x) * Heaviside(x - 1),
-log(x) * Heaviside(-x + 1) + log(1 - 1 / x) * Heaviside(x - 1) +
log(-x + 1) * Heaviside(-x + 1),
]
# 8.4.14
assert IMT(-gamma(s + Rational(1, 2))/(sqrt(pi)*s), s, x, (-Rational(1, 2), 0)) == \
erf(sqrt(x))
# 8.4.19
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, Rational(3, 4)))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(Rational(1, 2) - 2*s)*gamma(s + (a + 1)/2)
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-(re(a) + 1)/2, Rational(1, 4)))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(Rational(1, 2) - 2*s)
/ (gamma(Rational(1, 2) - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, Rational(1, 4)))) == \
cos(sqrt(x))*besselj(a, sqrt(x))
# TODO this comes out as an amazing mess, but simplifies nicely
assert simplify(IMT(gamma(a + s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), Rational(1, 2)))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(Rational(1, 2) - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, Rational(1, 2)))) == \
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(4**s*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)),
s, x, (-(re(a) + re(b))/2, Rational(1, 2)))) == \
besselj(a, sqrt(x))*besselj(b, sqrt(x))
# Section 8.4.20
# TODO this can be further simplified!
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
(pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
s, x,
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), Rational(1, 2)))) == \
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
# TODO more
# for coverage
assert IMT(pi / cos(pi * s), s, x,
(0, Rational(1, 2))) == sqrt(x) / (x + 1)
def test_Function():
assert mathematica_code(f(x, y, z)) == 'f[x, y, z]'
assert mathematica_code(sin(x)**cos(x)) == 'Sin[x]^Cos[x]'
assert mathematica_code(sign(x)) == 'Sign[x]'
assert mathematica_code(atanh(x),
user_functions={'atanh':
'ArcTanh'}) == 'ArcTanh[x]'
assert (mathematica_code(meijerg(
((1, 1), (3, 4)), ((1, ), ()),
x)) == 'MeijerG[{{1, 1}, {3, 4}}, {{1}, {}}, x]')
assert (mathematica_code(hyper(
(1, 2, 3), (3, 4), x)) == 'HypergeometricPFQ[{1, 2, 3}, {3, 4}, x]')
assert mathematica_code(Min(x, y)) == 'Min[x, y]'
assert mathematica_code(Max(x, y)) == 'Max[x, y]'
assert mathematica_code(Max(x,
2)) == 'Max[2, x]' # issue sympy/sympy#15344
assert mathematica_code(binomial(x, y)) == 'Binomial[x, y]'
assert mathematica_code(log(x)) == 'Log[x]'
assert mathematica_code(tan(x)) == 'Tan[x]'
assert mathematica_code(cot(x)) == 'Cot[x]'
assert mathematica_code(asin(x)) == 'ArcSin[x]'
assert mathematica_code(acos(x)) == 'ArcCos[x]'
assert mathematica_code(atan(x)) == 'ArcTan[x]'
assert mathematica_code(acot(x)) == 'ArcCot[x]'
assert mathematica_code(sinh(x)) == 'Sinh[x]'
assert mathematica_code(cosh(x)) == 'Cosh[x]'
assert mathematica_code(tanh(x)) == 'Tanh[x]'
assert mathematica_code(coth(x)) == 'Coth[x]'
assert mathematica_code(asinh(x)) == 'ArcSinh[x]'
assert mathematica_code(acosh(x)) == 'ArcCosh[x]'
assert mathematica_code(atanh(x)) == 'ArcTanh[x]'
assert mathematica_code(acoth(x)) == 'ArcCoth[x]'
assert mathematica_code(sech(x)) == 'Sech[x]'
assert mathematica_code(csch(x)) == 'Csch[x]'
assert mathematica_code(erf(x)) == 'Erf[x]'
assert mathematica_code(erfi(x)) == 'Erfi[x]'
assert mathematica_code(erfc(x)) == 'Erfc[x]'
assert mathematica_code(conjugate(x)) == 'Conjugate[x]'
assert mathematica_code(re(x)) == 'Re[x]'
assert mathematica_code(im(x)) == 'Im[x]'
assert mathematica_code(polygamma(x, y)) == 'PolyGamma[x, y]'
assert mathematica_code(factorial(x)) == 'Factorial[x]'
assert mathematica_code(factorial2(x)) == 'Factorial2[x]'
assert mathematica_code(rf(x, y)) == 'Pochhammer[x, y]'
assert mathematica_code(gamma(x)) == 'Gamma[x]'
assert mathematica_code(zeta(x)) == 'Zeta[x]'
assert mathematica_code(Heaviside(x)) == 'UnitStep[x]'
assert mathematica_code(fibonacci(x)) == 'Fibonacci[x]'
assert mathematica_code(polylog(x, y)) == 'PolyLog[x, y]'
assert mathematica_code(loggamma(x)) == 'LogGamma[x]'
assert mathematica_code(uppergamma(x, y)) == 'Gamma[x, y]'
class MyFunc1(Function):
@classmethod
def eval(cls, x):
pass
class MyFunc2(Function):
@classmethod
def eval(cls, x, y):
pass
pytest.raises(
ValueError,
lambda: mathematica_code(MyFunc1(x),
user_functions={'MyFunc1': ['Myfunc1']}))
assert mathematica_code(MyFunc1(x),
user_functions={'MyFunc1':
'Myfunc1'}) == 'Myfunc1[x]'
assert mathematica_code(
MyFunc2(x, y),
user_functions={'MyFunc2':
[(lambda *x: False, 'Myfunc2')]}) == 'MyFunc2[x, y]'
def test_Min():
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
nn_ = Symbol('nn_', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
np = Symbol('np', nonpositive=True)
np_ = Symbol('np_', nonpositive=True)
assert Min(5, 4) == 4
assert Min(-oo, -oo) == -oo
assert Min(-oo, n) == -oo
assert Min(n, -oo) == -oo
assert Min(-oo, np) == -oo
assert Min(np, -oo) == -oo
assert Min(-oo, 0) == -oo
assert Min(0, -oo) == -oo
assert Min(-oo, nn) == -oo
assert Min(nn, -oo) == -oo
assert Min(-oo, p) == -oo
assert Min(p, -oo) == -oo
assert Min(-oo, oo) == -oo
assert Min(oo, -oo) == -oo
assert Min(n, n) == n
assert Min(n, np) == Min(n, np)
assert Min(np, n) == Min(np, n)
assert Min(n, 0) == n
assert Min(0, n) == n
assert Min(n, nn) == n
assert Min(nn, n) == n
assert Min(n, p) == n
assert Min(p, n) == n
assert Min(n, oo) == n
assert Min(oo, n) == n
assert Min(np, np) == np
assert Min(np, 0) == np
assert Min(0, np) == np
assert Min(np, nn) == np
assert Min(nn, np) == np
assert Min(np, p) == np
assert Min(p, np) == np
assert Min(np, oo) == np
assert Min(oo, np) == np
assert Min(0, 0) == 0
assert Min(0, nn) == 0
assert Min(nn, 0) == 0
assert Min(0, p) == 0
assert Min(p, 0) == 0
assert Min(0, oo) == 0
assert Min(oo, 0) == 0
assert Min(nn, nn) == nn
assert Min(nn, p) == Min(nn, p)
assert Min(p, nn) == Min(p, nn)
assert Min(nn, oo) == nn
assert Min(oo, nn) == nn
assert Min(p, p) == p
assert Min(p, oo) == p
assert Min(oo, p) == p
assert Min(oo, oo) == oo
assert isinstance(Min(n, n_), Min)
assert isinstance(Min(nn, nn_), Min)
assert isinstance(Min(np, np_), Min)
assert isinstance(Min(p, p_), Min)
# lists
pytest.raises(ValueError, lambda: Min())
assert Min(x, y) == Min(y, x)
assert Min(x, y, z) == Min(z, y, x)
assert Min(x, Min(y, z)) == Min(z, y, x)
assert Min(x, Max(y, -oo)) == Min(x, y)
assert Min(p, oo, n, p, p, p_) == n
assert Min(p_, n_, p) == n_
assert Min(n, oo, -7, p, p, 2) == Min(n, -7)
assert Min(2, x, p, n, oo, n_, p, 2, -2, -2) == Min(-2, x, n, n_)
assert Min(0, x, 1, y) == Min(0, x, y)
assert Min(1000, 100, -100, x, p, n) == Min(n, x, -100)
assert Min(cos(x), sin(x)) == Min(cos(x), sin(x))
assert Min(cos(x), sin(x)).subs({x: 1}) == cos(1)
assert Min(cos(x), sin(x)).subs({x: Rational(1, 2)}) == sin(Rational(1, 2))
pytest.raises(ValueError, lambda: Min(cos(x), sin(x)).subs({x: I}))
pytest.raises(ValueError, lambda: Min(I))
pytest.raises(ValueError, lambda: Min(I, x))
pytest.raises(ValueError, lambda: Min(zoo, x))
assert Min(1, x).diff(x) == Heaviside(1 - x)
assert Min(x, 1).diff(x) == Heaviside(1 - x)
assert Min(0, -x, 1 - 2*x).diff(x) == -Heaviside(x + Min(0, -2*x + 1)) \
- 2*Heaviside(2*x + Min(0, -x) - 1)
pytest.raises(ArgumentIndexError, lambda: Min(1, x).fdiff(3))
a, b = Symbol('a', extended_real=True), Symbol('b', extended_real=True)
# a and b are both real, Min(a, b) should be real
assert Min(a, b).is_extended_real
# issue sympy/sympy#7619
f = Function('f')
assert Min(1, 2 * Min(f(1), 2)) # doesn't fail
# issue sympy/sympy#7233
e = Min(0, x)
assert e.evalf == e.n
assert e.evalf(strict=False).args == (0, x)
def test_heaviside():
x, y = symbols('x, y', extended_real=True)
assert Heaviside(0) == 0.5
assert Heaviside(-5) == 0
assert Heaviside(1) == 1
assert Heaviside(nan) == nan
assert Heaviside(x).is_real
assert Heaviside(z).is_real is None
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(z + I).is_Function is True
assert Heaviside(I*z).is_Function is True
pytest.raises(ArgumentIndexError, lambda: Heaviside(x).fdiff(2))
pytest.raises(ValueError, lambda: Heaviside(I))
pytest.raises(ValueError, lambda: Heaviside(2 + 3*I))
def test_deltaintegrate():
assert deltaintegrate(x, x) is None
assert deltaintegrate(x + DiracDelta(x), x) is None
assert deltaintegrate(DiracDelta(x, 0), x) == Heaviside(x)
for n in range(10):
assert deltaintegrate(DiracDelta(x, n + 1), x) == DiracDelta(x, n)
assert deltaintegrate(DiracDelta(x), x) == Heaviside(x)
assert deltaintegrate(DiracDelta(-x), x) == Heaviside(x)
assert deltaintegrate(DiracDelta(x - y), x) == Heaviside(x - y)
assert deltaintegrate(DiracDelta(y - x), x) == Heaviside(x - y)
assert deltaintegrate(x * DiracDelta(x), x) == 0
assert deltaintegrate((x - y) * DiracDelta(x - y), x) == 0
assert deltaintegrate(DiracDelta(x)**2, x) == DiracDelta(0) * Heaviside(x)
assert deltaintegrate(y*DiracDelta(x)**2, x) == \
y*DiracDelta(0)*Heaviside(x)
assert deltaintegrate(DiracDelta(x, 1)**2, x) is None
assert deltaintegrate(y * DiracDelta(x, 1)**2, x) is None
assert deltaintegrate(DiracDelta(x) * f(x), x) == f(0) * Heaviside(x)
assert deltaintegrate(DiracDelta(-x) * f(x), x) == f(0) * Heaviside(x)
assert deltaintegrate(DiracDelta(x - 1) * f(x),
x) == f(1) * Heaviside(x - 1)
assert deltaintegrate(DiracDelta(1 - x) * f(x),
x) == f(1) * Heaviside(x - 1)
assert deltaintegrate(DiracDelta(x**2 + x - 2), x) == \
Heaviside(x - 1)/3 + Heaviside(x + 2)/3
p = cos(x) * (DiracDelta(x) + DiracDelta(x**2 - 1)) * sin(x) * (x - pi)
assert deltaintegrate(
p, x) - (-pi * (cos(1) * Heaviside(-1 + x) * sin(1) / 2 -
cos(1) * Heaviside(1 + x) * sin(1) / 2) +
cos(1) * Heaviside(1 + x) * sin(1) / 2 +
cos(1) * Heaviside(-1 + x) * sin(1) / 2) == 0
p = x_2 * DiracDelta(x - x_2) * DiracDelta(x_2 - x_1)
assert deltaintegrate(p,
x_2) == x * DiracDelta(x - x_1) * Heaviside(x_2 - x)
p = x * y**2 * z * DiracDelta(y - x) * DiracDelta(y - z) * DiracDelta(x -
z)
assert deltaintegrate(
p, y) == x**3 * z * DiracDelta(x - z)**2 * Heaviside(y - x)
assert deltaintegrate((x + 1) * DiracDelta(2 * x),
x) == Rational(1, 2) * Heaviside(x)
assert deltaintegrate((x + 1)*DiracDelta(2*x/3 + 4/Integer(9)), x) == \
Rational(1, 2) * Heaviside(x + Rational(2, 3))
a, b, c = symbols('a b c', commutative=False)
assert deltaintegrate(DiracDelta(x - y)*f(x - b)*f(x - a), x) == \
f(y - b)*f(y - a)*Heaviside(x - y)
p = f(x - a) * DiracDelta(x - y) * f(x - c) * f(x - b)
assert deltaintegrate(
p, x) == f(y - a) * f(y - c) * f(y - b) * Heaviside(x - y)
p = DiracDelta(x - z) * f(x - b) * f(x - a) * DiracDelta(x - y)
assert deltaintegrate(p, x) == DiracDelta(y - z)*f(y - b)*f(y - a) * \
Heaviside(x - y)
def test_mellin_transform():
from diofant import Max, Min
MT = mellin_transform
bpos = symbols('b', positive=True)
# 8.4.2
assert MT(x**nu*Heaviside(x - 1), x, s) == \
(-1/(nu + s), (-oo, -re(nu)), True)
assert MT(x**nu*Heaviside(1 - x), x, s) == \
(1/(nu + s), (-re(nu), oo), True)
assert MT((1 - x)**(beta - 1)*Heaviside(1 - x), x, s) == \
(gamma(beta)*gamma(s)/gamma(beta + s), (0, oo), re(-beta) < 0)
assert MT((x - 1)**(beta - 1)*Heaviside(x - 1), x, s) == \
(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
(-oo, -re(beta) + 1), re(-beta) < 0)
assert MT((1 + x)**(-rho), x, s) == \
(gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
# TODO also the conditions should be simplified
assert MT(abs(1 - x)**(-rho), x,
s) == (cos(pi * (rho / 2 - s)) * gamma(s) * gamma(rho - s) /
(cos(pi * rho / 2) * gamma(rho)), (0, re(rho)),
And(re(rho) - 1 < 0,
re(rho) < 1))
mt = MT((1 - x)**(beta - 1) * Heaviside(1 - x) + a *
(x - 1)**(beta - 1) * Heaviside(x - 1), x, s)
assert mt[1], mt[2] == ((0, -re(beta) + 1), True)
assert MT((x**a - b**a)/(x - b), x, s)[0] == \
pi*b**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s)))
assert MT((x**a - bpos**a)/(x - bpos), x, s) == \
(pi*bpos**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s))),
(Max(-re(a), 0), Min(1 - re(a), 1)), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, bpos), x, s) == \
(-a*(2*bpos)**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
(0, -re(a)/2), True)
expr = (sqrt(x + b**2) + b)**a / sqrt(x + b**2)
assert MT(expr.subs(b, bpos), x, s) == \
(2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(s)
* gamma(1 - a - 2*s)/gamma(1 - a - s),
(0, -re(a)/2 + Rational(1, 2)), True)
# 8.4.2
assert MT(exp(-x), x, s) == (gamma(s), (0, oo), True)
assert MT(exp(-1 / x), x, s) == (gamma(-s), (-oo, 0), True)
# 8.4.5
assert MT(log(x)**4 * Heaviside(1 - x), x, s) == (24 / s**5, (0, oo), True)
assert MT(log(x)**3 * Heaviside(x - 1), x, s) == (6 / s**4, (-oo, 0), True)
assert MT(log(x + 1), x, s) == (pi / (s * sin(pi * s)), (-1, 0), True)
assert MT(log(1 / x + 1), x, s) == (pi / (s * sin(pi * s)), (0, 1), True)
assert MT(log(abs(1 - x)), x, s) == (pi / (s * tan(pi * s)), (-1, 0), True)
assert MT(log(abs(1 - 1 / x)), x,
s) == (pi / (s * tan(pi * s)), (0, 1), True)
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x) / (x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2 / (x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x) / (x + 1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + Rational(1, 2))/(sqrt(pi)*s), (-Rational(1, 2), 0), True)
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3*I) == I
assert sign(-3*I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2*I).doit() == sqrt(2)*(2 + 2*I)/4
assert sign(2 + 3*I).simplify() == sign(2 + 3*I)
assert sign(2 + 2*I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_finite is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2*x) == sign(x)
assert sign(2*x) == sign(x)
assert sign(I*x) == I*sign(x)
assert sign(-2*I*x) == -I*sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2*p*x) == sign(x)
assert sign(n*x) == -sign(x)
assert sign(n*m*x) == sign(x)
x = Symbol('x', imaginary=True)
xn = Symbol('xn', imaginary=True, nonzero=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(-I*x)
assert sign(xn).doit() == xn / Abs(xn)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', extended_real=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
assert sign(sin(x)).nseries(x) == 1
y = Symbol('y')
assert sign(x*y).nseries(x).removeO() == sign(y)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is True
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(Abs(x)) == 0
assert Abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_extended_real is True
assert sign(nz).is_zero is False
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', extended_real=True)).is_nonnegative is None
assert sign(Symbol('x', extended_real=True)).is_nonpositive is None
assert sign(Symbol('x', extended_real=True, zero=False)).is_nonpositive is None
x, y = Symbol('x', extended_real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
assert sign(x).rewrite(Heaviside) == 2*Heaviside(x)-1
assert sign(y).rewrite(Heaviside) == sign(y)
# evaluate what can be evaluated
assert sign(exp_polar(I*pi)*pi) is Integer(-1)
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq) == 0
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = cbrt(expand(q**3))
d = p - q
assert sign(d) == 0
assert abs(sign(z)) == Abs(sign(z), evaluate=False)
def test_rewrite_MaxMin_as_Heaviside():
assert Max(0, x).rewrite(Heaviside) == x * Heaviside(x)
assert Max(3, x).rewrite(Heaviside) == x*Heaviside(x - 3) + \
3*Heaviside(-x + 3)
assert Max(0, x+2, 2*x).rewrite(Heaviside) == \
2*x*Heaviside(2*x)*Heaviside(x - 2) + \
(x + 2)*Heaviside(-x + 2)*Heaviside(x + 2)
assert Min(0, x).rewrite(Heaviside) == x * Heaviside(-x)
assert Min(3, x).rewrite(Heaviside) == x*Heaviside(-x + 3) + \
3*Heaviside(x - 3)
assert Min(x, -x, -2).rewrite(Heaviside) == \
x*Heaviside(-2*x)*Heaviside(-x - 2) - \
x*Heaviside(2*x)*Heaviside(x - 2) \
- 2*Heaviside(-x + 2)*Heaviside(x + 2)
def test_atan2():
assert atan2.nargs == FiniteSet(2)
assert atan2(0, 0) == nan
assert atan2(0, 1) == 0
assert atan2(1, 1) == pi/4
assert atan2(1, 0) == pi/2
assert atan2(1, -1) == 3*pi/4
assert atan2(0, -1) == pi
assert atan2(-1, -1) == -3*pi/4
assert atan2(-1, 0) == -pi/2
assert atan2(-1, 1) == -pi/4
i = symbols('i', imaginary=True)
r = symbols('r', extended_real=True)
eq = atan2(r, i)
ans = -I*log((i + I*r)/sqrt(i**2 + r**2))
reps = ((r, 2), (i, I))
assert eq.subs(reps) == ans.subs(reps)
x = Symbol('x', negative=True)
y = Symbol('y', negative=True)
assert atan2(y, x) == atan(y/x) - pi
y = Symbol('y', nonnegative=True)
assert atan2(y, x) == atan(y/x) + pi
y = Symbol('y')
assert atan2(y, x) == atan2(y, x, evaluate=False)
u = Symbol('u', positive=True)
assert atan2(0, u) == 0
u = Symbol('u', negative=True)
assert atan2(0, u) == pi
assert atan2(y, oo) == 0
assert atan2(y, -oo) == 2*pi*Heaviside(re(y)) - pi
assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
assert atan2(y, x).rewrite(atan) == 2*atan(y/(x + sqrt(x**2 + y**2)))
ex = atan2(y, x) - arg(x + I*y)
assert ex.subs({x: 2, y: 3}).rewrite(arg) == 0
assert ex.subs({x: 2, y: 3*I}).rewrite(arg) == -pi - I*log(sqrt(5)*I/5)
assert ex.subs({x: 2*I, y: 3}).rewrite(arg) == -pi/2 - I*log(sqrt(5)*I)
assert ex.subs({x: 2*I, y: 3*I}).rewrite(arg) == -pi + atan(Rational(2, 3)) + atan(Rational(3, 2))
i = symbols('i', imaginary=True)
r = symbols('r', extended_real=True)
e = atan2(i, r)
rewrite = e.rewrite(arg)
reps = {i: I, r: -2}
assert rewrite == -I*log(abs(I*i + r)/sqrt(abs(i**2 + r**2))) + arg((I*i + r)/sqrt(i**2 + r**2))
assert (e - rewrite).subs(reps).equals(0)
r1 = Symbol('r1', real=True, nonzero=True)
r2 = Symbol('r2', real=True, nonzero=True)
assert atan2(r1, r2).is_real
assert atan2(0, r1) == pi*(-Heaviside(r1) + 1)
r1 = Symbol('r1', real=True)
r2 = Symbol('r2', real=True)
assert atan2(r1, r2).is_real is None
assert atan2(r1, r2).rewrite(arg) == arg(I*r1 + r2)
assert conjugate(atan2(x, y)) == atan2(conjugate(x), conjugate(y))
assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
assert diff(atan2(y, x), y) == x/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2)
pytest.raises(ArgumentIndexError, lambda: atan2(x, y).fdiff(3))