def test_issue_841():
a, b, c, d = symbols('a:d', positive=True, bounded=True)
assert integrate(
exp(-x**2 + I*c*x), x) == sqrt(pi)*erf(x - I*c/2)*exp(-c**S(2)/4)/2
assert integrate(exp(a*x**2 + b*x + c), x) == \
I*sqrt(pi)*erf(-I*x*sqrt(
a) - I*b/(2*sqrt(a)))*exp(c)*exp(-b**2/(4*a))/(2*sqrt(a))
def test_branch_bug():
assert hyperexpand(hyper((-S(1)/3, S(1)/2), (S(2)/3, S(3)/2), -z)) == \
-z**S('1/3')*lowergamma(exp_polar(I*pi)/3, z)/5 \
+ sqrt(pi)*erf(sqrt(z))/(5*sqrt(z))
assert hyperexpand(meijerg([S(7)/6, 1], [], [S(2)/3], [S(1)/6, 0], z)) == \
2*z**S('2/3')*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) - 2*lowergamma(
S(2)/3, z)/z**S('2/3'))*gamma(S(2)/3)/gamma(S(5)/3)
def test_to_expr():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand()
q = 1/(x**2 - 2*x + 1)
assert p == q
p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr()
q = (sin(x)**2/x).integrate((x, 0, x))
assert p == q
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
p = expr_to_holonomic(log(1+x**2)).to_expr()
q = C_2*log(x**2 + 1)
assert p == q
p = expr_to_holonomic(log(1+x**2)).diff().to_expr()
q = C_0*x/(x**2 + 1)
assert p == q
p = expr_to_holonomic(erf(x) + x).to_expr()
q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi)
assert p == q
p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
assert p == sqrt(x)
assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x)
p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
assert p == sqrt(1+x**2)
p = expr_to_holonomic((2*x**2 + 1)**(S(2)/3)).to_expr()
assert p == (2*x**2 + 1)**(S(2)/3)
p = expr_to_holonomic(sqrt(-x**2+2*x)).to_expr()
assert p == sqrt(x)*sqrt(-x + 2)
p = expr_to_holonomic((-2*x**3+7*x)**(S(2)/3)).to_expr()
q = x**(S(2)/3)*(-2*x**2 + 7)**(S(2)/3)
assert p == q
p = from_hyper(hyper((-2, -3), (S(1)/2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1)/2, ), x))
D_0 = Symbol('D_0')
C_0 = Symbol('C_0')
assert (p.to_expr().subs({C_0:1, D_0:0}) - s).simplify() == 0
p.y0 = {0: [1], S(1)/2: [0]}
assert p.to_expr() == s
assert expr_to_holonomic(x**5).to_expr() == x**5
assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \
2*x**3-3*x**2
a = symbols("a")
p = (expr_to_holonomic(1.4*x)*expr_to_holonomic(a*x, x)).to_expr()
q = 1.4*a*x**2
assert p == q
p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(a*x, x)).to_expr()
q = x*(a + 1.4)
assert p == q
p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(x)).to_expr()
assert p == 2.4*x
def test_manualintegrate_special():
f, F = 4*exp(-x**2/3), 2*sqrt(3)*sqrt(pi)*erf(sqrt(3)*x/3)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 3*exp(4*x**2), 3*sqrt(pi)*erfi(2*x)/4
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = x**(S(1)/3)*exp(-x/8), -16*uppergamma(S(4)/3, x/8)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = exp(2*x)/x, Ei(2*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = exp(1 + 2*x - x**2), sqrt(pi)*exp(2)*erf(x - 1)/2
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f = sin(x**2 + 4*x + 1)
F = (sqrt(2)*sqrt(pi)*(-sin(3)*fresnelc(sqrt(2)*(2*x + 4)/(2*sqrt(pi))) +
cos(3)*fresnels(sqrt(2)*(2*x + 4)/(2*sqrt(pi))))/2)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = cos(4*x**2), sqrt(2)*sqrt(pi)*fresnelc(2*sqrt(2)*x/sqrt(pi))/4
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = sin(3*x + 2)/x, sin(2)*Ci(3*x) + cos(2)*Si(3*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = sinh(3*x - 2)/x, -sinh(2)*Chi(3*x) + cosh(2)*Shi(3*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 5*cos(2*x - 3)/x, 5*cos(3)*Ci(2*x) + 5*sin(3)*Si(2*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = cosh(x/2)/x, Chi(x/2)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = cos(x**2)/x, Ci(x**2)/2
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 1/log(2*x + 1), li(2*x + 1)/2
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = polylog(2, 5*x)/x, polylog(3, 5*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 5/sqrt(3 - 2*sin(x)**2), 5*sqrt(3)*elliptic_f(x, S(2)/3)/3
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = sqrt(4 + 9*sin(x)**2), 2*elliptic_e(x, -S(9)/4)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
def test_to_expr():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, 1).to_expr()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr()
q = 1/(x**2 - 2*x + 1)
assert p == q
p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr()
q = (sin(x)**2/x).integrate((x, 0, x))
assert p == q
C_1, C_2, C_3 = symbols('C_1, C_2, C_3')
p = expr_to_holonomic(log(1+x**2)).to_expr()
q = C_2*log(x**2 + 1)
assert p == q
p = expr_to_holonomic(log(1+x**2)).diff().to_expr()
q = C_1*x/(x**2 + 1)
assert p == q
p = expr_to_holonomic(erf(x) + x).to_expr()
q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi)
assert p == q
p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
assert p == sqrt(x)
p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
assert p == sqrt(1+x**2)
p = expr_to_holonomic((2*x**2 + 1)**(S(2)/3)).to_expr()
assert p == (2*x**2 + 1)**(S(2)/3)
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_simplify_other():
assert simplify(sin(x) ** 2 + cos(x) ** 2) == 1
assert simplify(gamma(x + 1) / gamma(x)) == x
assert simplify(sin(x) ** 2 + cos(x) ** 2 + factorial(x) / gamma(x)) == 1 + x
assert simplify(Eq(sin(x) ** 2 + cos(x) ** 2, factorial(x) / gamma(x))) == Eq(x, 1)
nc = symbols("nc", commutative=False)
assert simplify(x + x * nc) == x * (1 + nc)
# issue 6123
# f = exp(-I*(k*sqrt(t) + x/(2*sqrt(t)))**2)
# ans = integrate(f, (k, -oo, oo), conds='none')
ans = I * (
-pi
* x
* exp(-3 * I * pi / 4 + I * x ** 2 / (4 * t))
* erf(x * exp(-3 * I * pi / 4) / (2 * sqrt(t)))
/ (2 * sqrt(t))
+ pi * x * exp(-3 * I * pi / 4 + I * x ** 2 / (4 * t)) / (2 * sqrt(t))
) * exp(-I * x ** 2 / (4 * t)) / (sqrt(pi) * x) - I * sqrt(pi) * (
-erf(x * exp(I * pi / 4) / (2 * sqrt(t))) + 1
) * exp(
I * pi / 4
) / (
2 * sqrt(t)
)
assert simplify(ans) == -(-1) ** (S(3) / 4) * sqrt(pi) / sqrt(t)
# issue 6370
assert simplify(2 ** (2 + x) / 4) == 2 ** x
def test_series():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 + 2*x*Dx, x, 0, [0, 1]).series(n=10)
q = x - x**3/3 + x**5/10 - x**7/42 + x**9/216 + O(x**10)
assert p == q
p = HolonomicFunction(Dx - 1, x).composition(x**2, 0, [1]) # e^(x**2)
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # cos(x)
r = (p * q).series(n=10) # expansion of cos(x) * exp(x**2)
s = 1 + x**2/2 + x**4/24 - 31*x**6/720 - 179*x**8/8064 + O(x**10)
assert r == s
t = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) # log(1 + x)
r = (p * t + q).series(n=10)
s = 1 + x - x**2 + 4*x**3/3 - 17*x**4/24 + 31*x**5/30 - 481*x**6/720 +\
71*x**7/105 - 20159*x**8/40320 + 379*x**9/840 + O(x**10)
assert r == s
p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
(4-6*x**3+2*x**4)*Dx**2, x, 0, [0, 1]).series(n=7)
q = x + x**3/6 - 3*x**4/16 + x**5/20 - 23*x**6/960 + O(x**7)
assert p == q
p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
(4-6*x**3+2*x**4)*Dx**2, x, 0, [1, 0]).series(n=7)
q = 1 - 3*x**2/4 - x**3/4 - 5*x**4/32 - 3*x**5/40 - 17*x**6/384 + O(x**7)
assert p == q
p = expr_to_holonomic(erf(x) + x).series(n=10)
C_3 = symbols('C_3')
q = (erf(x) + x).series(n=10)
assert p.subs(C_3, -2/(3*sqrt(pi))) == q
assert expr_to_holonomic(sqrt(x**3 + x)).series(n=10) == sqrt(x**3 + x).series(n=10)
assert expr_to_holonomic((2*x - 3*x**2)**(S(1)/3)).series() == ((2*x - 3*x**2)**(S(1)/3)).series()
assert expr_to_holonomic(sqrt(x**2-x)).series() == (sqrt(x**2-x)).series()
assert expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).series(n=10) == (cos(x)**2/x**2).series(n=10)
assert expr_to_holonomic(cos(x)**2/x**2, x0=1).series(n=10) == (cos(x)**2/x**2).series(n=10, x0=1)
assert expr_to_holonomic(cos(x-1)**2/(x-1)**2, x0=1, y0={-2: [1, 0, -1]}).series(n=10) \
== (cos(x-1)**2/(x-1)**2).series(x0=1, n=10)
def test_issue_1791():
z = Symbol("z", positive=True)
assert integrate(exp(-log(x) ** 2), x) == pi ** (S(1) / 2) * erf(-S(1) / 2 + log(x)) * exp(S(1) / 4) / 2
assert integrate(exp(log(x) ** 2), x) == -I * pi ** (S(1) / 2) * erf(I * log(x) + I / 2) * exp(-S(1) / 4) / 2
assert integrate(exp(-z * log(x) ** 2), x) == pi ** (S(1) / 2) * erf(
z ** (S(1) / 2) * log(x) - 1 / (2 * z ** (S(1) / 2))
) * exp(S(1) / (4 * z)) / (2 * z ** (S(1) / 2))
def test_errorinverses():
assert solveset_real(erf(x) - S.One/2, x) == \
FiniteSet(erfinv(S.One/2))
assert solveset_real(erfinv(x) - 2, x) == \
FiniteSet(erf(2))
assert solveset_real(erfc(x) - S.One, x) == \
FiniteSet(erfcinv(S.One))
assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2))
def test_branch_bug():
from sympy import powdenest, lowergamma
# TODO combsimp cannot prove that the factor is unity
assert powdenest(integrate(erf(x**3), x, meijerg=True).diff(x),
polar=True) == 2*erf(x**3)*gamma(S(2)/3)/3/gamma(S(5)/3)
assert integrate(erf(x**3), x, meijerg=True) == \
2*x*erf(x**3)*gamma(S(2)/3)/(3*gamma(S(5)/3)) \
- 2*gamma(S(2)/3)*lowergamma(S(2)/3, x**6)/(3*sqrt(pi)*gamma(S(5)/3))
def test_issue_1791():
z = Symbol('z', positive=True)
assert integrate(exp(-log(x)**2), x) == \
sqrt(pi)*erf(-S(1)/2 + log(x))*exp(S(1)/4)/2
assert integrate(exp(log(x)**2), x) == \
-I*sqrt(pi)*exp(-S(1)/4)*erf(I*log(x) + I/2)/2
assert integrate(exp(-z*log(x)**2), x) == sqrt(pi)*erf(sqrt(z)*log(x)
- 1/(2*sqrt(z)))*exp(S(1)/(4*z))/(2*sqrt(z))
def test_issue_4890():
z = Symbol('z', positive=True)
assert integrate(exp(-log(x)**2), x) == \
sqrt(pi)*exp(S(1)/4)*erf(log(x)-S(1)/2)/2
assert integrate(exp(log(x)**2), x) == \
sqrt(pi)*exp(-S(1)/4)*erfi(log(x)+S(1)/2)/2
assert integrate(exp(-z*log(x)**2), x) == \
sqrt(pi)*exp(1/(4*z))*erf(sqrt(z)*log(x) - 1/(2*sqrt(z)))/(2*sqrt(z))
def fdiff(self, argindex=2):
if argindex == 2:
x0, x1 = self.args
return -2 * exp(-x1 ** 2) / (sqrt(pi) * (erf(x0) - erf(x1)))
elif argindex == 1:
x0, x1 = self.args
return 2.0 * exp(-x0 ** 2) / (sqrt(pi) * (erf(x0) - erf(x1)))
else:
raise ArgumentIndexError(self, argindex)
def test_issue_841():
a,b,c,d = symbols('a:d', positive=True, bounded=True)
assert integrate(exp(-x**2 + I*c*x), x) == sqrt(pi)*erf(x - I*c/2)*exp(-c**S(2)/4)/2
assert integrate(exp(a*x**2 + b*x + c), x) == I*sqrt(pi)*erf(-I*x*sqrt(a) - I*b/(2*sqrt(a)))*exp(c)*exp(-b**2/(4*a))/(2*sqrt(a))
a,b,c,d = symbols('a:d', positive=True)
i = integrate(exp(-a*x**2 + 2*d*x), (x, -oo, oo))
ans = sqrt(pi)*exp(d**2/a)*(1 + erf(oo - d/sqrt(a)))/(2*sqrt(a))
n, d = i.as_numer_denom()
assert factor(n, expand=False)/d == ans
def test_erfinv():
assert erfinv(0) == 0
assert erfinv(1) == S.Infinity
assert erfinv(nan) == S.NaN
assert erfinv(erf(w)) == w
assert erfinv(erf(-w)) == -w
assert erfinv(x).diff() == sqrt(pi)*exp(erfinv(x)**2)/2
assert erfinv(z).rewrite('erfcinv') == erfcinv(1-z)
def test_issue_841():
from sympy import simplify
a = Symbol('a', positive = True)
b = Symbol('b')
c = Symbol('c')
d = Symbol('d', positive = True)
assert integrate(exp(-x**2 + I*c*x), x) == pi**(S(1)/2)*erf(x - I*c/2)*exp(-c**S(2)/4)/2
assert integrate(exp(a*x**2 + b*x + c), x) == \
I*pi**(S(1)/2)*erf(-I*x*a**(S(1)/2) - I*b/(2*a**(S(1)/2)))*exp(c)*exp(-b**2/(4*a))/(2*a**(S(1)/2))
assert simplify(integrate(exp(-a*x**2 + 2*d*x), (x, -oo, oo))) == pi**(S(1)/2)*(1 + erf(oo - d/a**(S(1)/2))) \
*exp(d**2/a)/(2*a**(S(1)/2))
def test_erf():
assert erf(nan) == nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) == 0
assert erf(-2) == -erf(2)
assert erf(-x*y) == -erf(x*y)
assert erf(-x - y) == -erf(x + y)
def test_laplace_transform():
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) + log(s) + EulerGamma)/(-s), 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) == (1/b/(1 + (a + s)**2/b**2), -a, True)
assert LT(exp(-a*t)*cos(b*t), t, s) == \
(1/(s + a)/(1 + b**2/(a + s)**2), -a, True)
# TODO sinh, cosh have delicate cancellation
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),
]
def test_recursive():
from sympy import symbols, exp_polar
a, b, c = symbols("a b c", positive=True)
assert simplify(integrate(exp(-(x - a) ** 2) * exp(-(x - b) ** 2), (x, 0, oo))) == sqrt(2 * pi) / 4 * (
1 + erf(sqrt(2) / 2 * (a + b))
) * exp(-a ** 2 - b ** 2 + (a + b) ** 2 / 2)
assert simplify(integrate(exp(-(x - a) ** 2) * exp(-(x - b) ** 2) * exp(c * x), (x, 0, oo))) == sqrt(2 * pi) / 4 * (
1 + erf(sqrt(2) / 4 * (2 * a + 2 * b + c))
) * exp(-a ** 2 - b ** 2 + (2 * a + 2 * b + c) ** 2 / 8)
assert simplify(integrate(exp(-(x - a - b - c) ** 2), (x, 0, oo))) == sqrt(pi) / 2 * (1 + erf(a + b + c))
assert simplify(integrate(exp(-(x + a + b + c) ** 2), (x, 0, oo))) == sqrt(pi) / 2 * (1 - erf(a + b + c))
def test_gruntz_eval_special():
# Gruntz, p. 126
assert gruntz(exp(x) * (sin(1 / x + exp(-x)) - sin(1 / x + exp(-x ** 2))), x, oo) == 1
assert gruntz((erf(x - exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x ** 2), x, oo) == -2 / sqrt(pi)
assert gruntz(exp(exp(x)) * (exp(sin(1 / x + exp(-exp(x)))) - exp(sin(1 / x))), x, oo) == 1
assert gruntz(exp(x) * (gamma(x + exp(-x)) - gamma(x)), x, oo) == oo
assert gruntz(exp(exp(digamma(digamma(x)))) / x, x, oo) == exp(-S(1) / 2)
assert gruntz(exp(exp(digamma(log(x)))) / x, x, oo) == exp(-S(1) / 2)
assert gruntz(digamma(digamma(digamma(x))), x, oo) == oo
assert gruntz(loggamma(loggamma(x)), x, oo) == oo
assert gruntz(((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x) - cos(1 / x)) * x * log(x), x, oo) == -S(1) / 2
assert gruntz(x * (gamma(x - 1 / gamma(x)) - gamma(x) + log(x)), x, oo) == S(1) / 2
assert gruntz((gamma(x + 1 / gamma(x)) - gamma(x)) / log(x), x, oo) == 1
def test_erf_f_code():
x = symbols('x')
routine = make_routine("test", erf(x) - erf(-2 * x))
code_gen = FCodeGen()
source = get_string(code_gen.dump_f95, [routine])
expected = (
"REAL*8 function test(x)\n"
"implicit none\n"
"REAL*8, intent(in) :: x\n"
"test = erf(x) + erf(2.0d0*x)\n"
"end function\n"
)
assert source == expected, source
def test_recursive():
from sympy import symbols, exp_polar, expand
a, b, c = symbols('a b c', positive=True)
assert simplify(integrate(exp(-(x - a)**2)*exp(-(x - b)**2), (x, 0, oo))) \
== (sqrt(2)*sqrt(pi)*(erf(sqrt(2)*(a + b)/2) + 1)*exp(
-a**2/2 + a*b - b**2/2))/4
assert simplify(integrate(
exp(-(x - a)**2)*exp(-(x - b)**2)*exp(c*x), (x, 0, oo))) == \
(sqrt(2)*sqrt(pi)*(erf(sqrt(2)*(2*a + 2*b + c)/4) + 1)*
exp(-a**2/2 + a*b + a*c/2 - b**2/2 + b*c/2 + c**2/8)/4)
assert simplify(integrate(exp(-(x - a - b - c)**2), (x, 0, oo))) == \
sqrt(pi)/2*(1 + erf(a + b + c))
assert simplify(integrate(exp(-(x + a + b + c)**2), (x, 0, oo))) == \
sqrt(pi)/2*(1 - erf(a + b + c))
def test_from_sympy():
x = symbols("x")
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), "Dx")
p = from_sympy((sin(x) / x) ** 2)
q = HolonomicFunction(
8 * x + (4 * x ** 2 + 6) * Dx + 6 * x * Dx ** 2 + x ** 2 * Dx ** 3,
x,
1,
[sin(1) ** 2, -2 * sin(1) ** 2 + 2 * sin(1) * cos(1), -8 * sin(1) * cos(1) + 2 * cos(1) ** 2 + 4 * sin(1) ** 2],
)
assert p == q
p = from_sympy(1 / (1 + x ** 2) ** 2)
q = HolonomicFunction(4 * x + (x ** 2 + 1) * Dx, x, 0, 1)
assert p == q
p = from_sympy(exp(x) * sin(x) + x * log(1 + x))
q = HolonomicFunction(
(4 * x ** 3 + 20 * x ** 2 + 40 * x + 36)
+ (-4 * x ** 4 - 20 * x ** 3 - 40 * x ** 2 - 36 * x) * Dx
+ (4 * x ** 5 + 12 * x ** 4 + 14 * x ** 3 + 16 * x ** 2 + 20 * x - 8) * Dx ** 2
+ (-4 * x ** 5 - 10 * x ** 4 - 4 * x ** 3 + 4 * x ** 2 - 2 * x + 8) * Dx ** 3
+ (2 * x ** 5 + 4 * x ** 4 - 2 * x ** 3 - 7 * x ** 2 + 2 * x + 5) * Dx ** 4,
x,
0,
[0, 1, 4, -1],
)
assert p == q
p = from_sympy(x * exp(x) + cos(x) + 1)
q = HolonomicFunction(
(-x - 3) * Dx + (x + 2) * Dx ** 2 + (-x - 3) * Dx ** 3 + (x + 2) * Dx ** 4, x, 0, [2, 1, 1, 3]
)
assert p == q
assert (x * exp(x) + cos(x) + 1).series(n=10) == p.series(n=10)
p = from_sympy(log(1 + x) ** 2 + 1)
q = HolonomicFunction(Dx + (3 * x + 3) * Dx ** 2 + (x ** 2 + 2 * x + 1) * Dx ** 3, x, 0, [1, 0, 2])
assert p == q
p = from_sympy(erf(x) ** 2 + x)
q = HolonomicFunction(
(32 * x ** 4 - 8 * x ** 2 + 8) * Dx ** 2 + (24 * x ** 3 - 2 * x) * Dx ** 3 + (4 * x ** 2 + 1) * Dx ** 4,
x,
0,
[0, 1, 8 / pi, 0],
)
assert p == q
p = from_sympy(cosh(x) * x)
q = HolonomicFunction((-x ** 2 + 2) - 2 * x * Dx + x ** 2 * Dx ** 2, x, 0, [0, 1])
assert p == q
p = from_sympy(besselj(2, x))
q = HolonomicFunction((x ** 2 - 4) + x * Dx + x ** 2 * Dx ** 2, x, 0, [0, 0])
assert p == q
p = from_sympy(besselj(0, x) + exp(x))
q = HolonomicFunction(
(-2 * x ** 2 - x + 1)
+ (2 * x ** 2 - x - 3) * Dx
+ (-2 * x ** 2 + x + 2) * Dx ** 2
+ (2 * x ** 2 + x) * Dx ** 3,
x,
0,
[2, 1, 1 / 2],
)
assert p == q
def test_lowergamma():
from sympy import meijerg, exp_polar, I, expint
assert lowergamma(x, y).diff(y) == y**(x-1)*exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert lowergamma(x, y).diff(x) == \
gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
+ meijerg([], [1, 1], [0, 0, x], [], y)
assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x))
assert not lowergamma(S.Half - 3, x).has(lowergamma)
assert not lowergamma(S.Half + 3, x).has(lowergamma)
assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
assert tn(lowergamma(S.Half + 3, x, evaluate=False),
lowergamma(S.Half + 3, x), x)
assert tn(lowergamma(S.Half - 3, x, evaluate=False),
lowergamma(S.Half - 3, x), x)
assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)
assert tn_branch(-3, lowergamma)
assert tn_branch(-4, lowergamma)
assert tn_branch(S(1)/3, lowergamma)
assert tn_branch(pi, lowergamma)
assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
assert lowergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
assert lowergamma(x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
k = Symbol('k', integer=True)
assert lowergamma(k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
k = Symbol('k', integer=True, positive=False)
assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
def test_uppergamma():
from sympy import meijerg, exp_polar, I, expint
assert uppergamma(4, 0) == 6
assert uppergamma(x, y).diff(y) == -y**(x-1)*exp(-y)
assert td(uppergamma(randcplx(), y), y)
assert uppergamma(x, y).diff(x) == \
uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
assert td(uppergamma(x, randcplx()), x)
assert uppergamma(S.Half, x) == sqrt(pi)*(1 - erf(sqrt(x)))
assert not uppergamma(S.Half - 3, x).has(uppergamma)
assert not uppergamma(S.Half + 3, x).has(uppergamma)
assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
assert tn(uppergamma(S.Half + 3, x, evaluate=False),
uppergamma(S.Half + 3, x), x)
assert tn(uppergamma(S.Half - 3, x, evaluate=False),
uppergamma(S.Half - 3, x), x)
assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)
assert tn_branch(-3, uppergamma)
assert tn_branch(-4, uppergamma)
assert tn_branch(S(1)/3, uppergamma)
assert tn_branch(pi, uppergamma)
assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x)
assert uppergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + gamma(y)*(1-exp(4*pi*I*y))
assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I
assert uppergamma(-2, x) == expint(3, x)/x**2
assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)
def test_recursive():
from sympy import symbols
a, b, c = symbols('a b c', positive=True)
r = exp(-(x - a)**2)*exp(-(x - b)**2)
e = integrate(r, (x, 0, oo), meijerg=True)
assert simplify(e.expand()) == (
sqrt(2)*sqrt(pi)*(
(erf(sqrt(2)*(a + b)/2) + 1)*exp(-a**2/2 + a*b - b**2/2))/4)
e = integrate(exp(-(x - a)**2)*exp(-(x - b)**2)*exp(c*x), (x, 0, oo), meijerg=True)
assert simplify(e) == (
sqrt(2)*sqrt(pi)*(erf(sqrt(2)*(2*a + 2*b + c)/4) + 1)*exp(-a**2 - b**2
+ (2*a + 2*b + c)**2/8)/4)
assert simplify(integrate(exp(-(x - a - b - c)**2), (x, 0, oo), meijerg=True)) == \
sqrt(pi)/2*(1 + erf(a + b + c))
assert simplify(integrate(exp(-(x + a + b + c)**2), (x, 0, oo), meijerg=True)) == \
sqrt(pi)/2*(1 - erf(a + b + c))
def test_to_Sequence_Initial_Coniditons():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
n = symbols('n', integer=True)
_, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
p = HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence()
q = [(HolonomicSequence(-1 + (n + 1)*Sn, 1), 0)]
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence()
q = [(HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1]), 0)]
assert p == q
p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence()
q = [(HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, -1/2, 1/12]), 1)]
assert p == q
p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence()
q = [(HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2), 0, 3)]
assert p == q
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
p = expr_to_holonomic(log(1+x**2))
q = [(HolonomicSequence(n**2 + (n**2 + 2*n)*Sn**2, [0, 0, C_2]), 0, 1)]
assert p.to_sequence() == q
p = p.diff()
q = [(HolonomicSequence((n + 2) + (n + 2)*Sn**2, [C_0, 0]), 1, 0)]
assert p.to_sequence() == q
p = expr_to_holonomic(erf(x) + x).to_sequence()
q = [(HolonomicSequence((2*n**2 - 2*n) + (n**3 + 2*n**2 - n - 2)*Sn**2, [0, 1 + 2/sqrt(pi), 0, C_3]), 0, 2)]
assert p == q
def test_NormalDistribution():
nd = NormalDistribution(0, 1)
x = Symbol('x')
assert nd.cdf(x) == erf(sqrt(2)*x/2)/2 + S.One/2
assert isinstance(nd.sample(), float) or nd.sample().is_Number
assert nd.expectation(1, x) == 1
assert nd.expectation(x, x) == 0
assert nd.expectation(x**2, x) == 1
def test_expr_to_holonomic():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic((sin(x)/x)**2)
q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \
[1, 0, -2/3])
assert p == q
p = expr_to_holonomic(1/(1+x**2)**2)
q = HolonomicFunction(4*x + (x**2 + 1)*Dx, x, 0, 1)
assert p == q
p = expr_to_holonomic(exp(x)*sin(x)+x*log(1+x))
q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \
- 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \
(-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \
7*x**2/2 + x + 5/2)*Dx**4, x, 0, [0, 1, 4, -1])
assert p == q
p = expr_to_holonomic(x*exp(x)+cos(x)+1)
q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \
0, [2, 1, 1, 3])
assert p == q
assert (x*exp(x)+cos(x)+1).series(n=10) == p.series(n=10)
p = expr_to_holonomic(log(1 + x)**2 + 1)
q = HolonomicFunction(Dx + (3*x + 3)*Dx**2 + (x**2 + 2*x + 1)*Dx**3, x, 0, [1, 0, 2])
assert p == q
p = expr_to_holonomic(erf(x)**2 + x)
q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \
(x**2+ 1/4)*Dx**4, x, 0, [0, 1, 8/pi, 0])
assert p == q
p = expr_to_holonomic(cosh(x)*x)
q = HolonomicFunction((-x**2 + 2) -2*x*Dx + x**2*Dx**2, x, 0, [0, 1])
assert p == q
p = expr_to_holonomic(besselj(2, x))
q = HolonomicFunction((x**2 - 4) + x*Dx + x**2*Dx**2, x, 0, [0, 0])
assert p == q
p = expr_to_holonomic(besselj(0, x) + exp(x))
q = HolonomicFunction((-x**2 - x/2 + 1/2) + (x**2 - x/2 - 3/2)*Dx + (-x**2 + x/2 + 1)*Dx**2 +\
(x**2 + x/2)*Dx**3, x, 0, [2, 1, 1/2])
assert p == q
p = expr_to_holonomic(sin(x)**2/x)
q = HolonomicFunction(4 + 4*x*Dx + 3*Dx**2 + x*Dx**3, x, 0, [0, 1, 0])
assert p == q
p = expr_to_holonomic(sin(x)**2/x, x0=2)
q = HolonomicFunction((4) + (4*x)*Dx + (3)*Dx**2 + (x)*Dx**3, x, 2, [sin(2)**2/2,
sin(2)*cos(2) - sin(2)**2/4, -3*sin(2)**2/4 + cos(2)**2 - sin(2)*cos(2)])
assert p == q
p = expr_to_holonomic(log(x)/2 - Ci(2*x)/2 + Ci(2)/2)
q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \
[-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0])
assert p == q
p = p.to_expr()
q = log(x)/2 - Ci(2*x)/2 + Ci(2)/2
assert p == q
p = expr_to_holonomic(x**(S(1)/2), x0=1)
q = HolonomicFunction(x*Dx - 1/2, x, 1, 1)
assert p == q
p = expr_to_holonomic(sqrt(1 + x**2))
q = HolonomicFunction((-x) + (x**2 + 1)*Dx, x, 0, 1)
assert p == q
def test_heurisch_special():
assert heurisch(erf(x), x) == x * erf(x) + exp(-x**2) / sqrt(pi)
assert heurisch(exp(-x**2) * erf(x), x) == sqrt(pi) * erf(x)**2 / 4
def test_erf():
assert erf(nan) == nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) == 0
assert erf(I * oo) == oo * I
assert erf(-I * oo) == -oo * I
assert erf(-2) == -erf(2)
assert erf(-x * y) == -erf(x * y)
assert erf(-x - y) == -erf(x + y)
assert erf(erfinv(x)) == x
assert erf(erfcinv(x)) == 1 - x
assert erf(erf2inv(0, x)) == x
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_real is False
assert erf(0).is_real is True
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi)
assert erf(1 / x).as_leading_term(x) == erf(1 / x)
assert erf(z).rewrite('uppergamma') == sqrt(z**2) * erf(sqrt(z**2)) / z
assert erf(z).rewrite('erfc') == S.One - erfc(z)
assert erf(z).rewrite('erfi') == -I * erfi(I * z)
assert erf(z).rewrite('fresnels') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('fresnelc') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('hyper') == 2 * z * hyper([S.Half], [3 * S.Half],
-z**2) / sqrt(pi)
assert erf(z).rewrite('meijerg') == z * meijerg([S.Half], [], [0],
[-S.Half], z**2) / sqrt(pi)
assert erf(z).rewrite(
'expint') == sqrt(z**2) / z - z * expint(S.Half, z**2) / sqrt(S.Pi)
assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
2/sqrt(pi)
assert limit((1 - erf(z)) * exp(z**2) * z, z, oo) == 1 / sqrt(pi)
assert limit((1 - erf(x)) * exp(x**2) * sqrt(pi) * x, x, oo) == 1
assert limit(((1 - erf(x)) * exp(x**2) * sqrt(pi) * x - 1) * 2 * x**2, x,
oo) == -1
assert erf(x).as_real_imag() == \
((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
def test_issue_3796():
assert manualintegrate(diff(exp(x + x**2)), x) == exp(x + x**2)
assert integrate(x * exp(x**4), x,
risch=False) == -I * sqrt(pi) * erf(I * x**2) / 4
def test_issue_8433():
d, t = symbols('d t', positive=True)
assert limit(erf(1 - t/d), t, oo) == -1
def test_issue_13750():
a = Symbol('a')
assert limit(erf(a - x), x, oo) == -1
assert limit(erf(sqrt(x) - x), x, oo) == -1
def test_checksysodesol():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq = (Eq(diff(x(t), t), 9 * y(t)), Eq(diff(y(t), t), 12 * x(t)))
sol = [Eq(x(t), 9*C1*exp(-6*sqrt(3)*t) + 9*C2*exp(6*sqrt(3)*t)), \
Eq(y(t), -6*sqrt(3)*C1*exp(-6*sqrt(3)*t) + 6*sqrt(3)*C2*exp(6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t),
2 * x(t) + 4 * y(t)), Eq(diff(y(t), t), 12 * x(t) + 41 * y(t)))
sol = [Eq(x(t), 4*C1*exp(t*(-sqrt(1713)/2 + Rational(43, 2))) + 4*C2*exp(t*(sqrt(1713)/2 + \
Rational(43, 2)))), Eq(y(t), C1*(-sqrt(1713)/2 + Rational(39, 2))*exp(t*(-sqrt(1713)/2 + \
Rational(43, 2))) + C2*(Rational(39, 2) + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + Rational(43, 2))))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t),
x(t) + y(t)), Eq(diff(y(t), t), -2 * x(t) + 2 * y(t)))
sol = [Eq(x(t), (C1*sin(sqrt(7)*t/2) + C2*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2))), \
Eq(y(t), ((C1/2 - sqrt(7)*C2/2)*sin(sqrt(7)*t/2) + (sqrt(7)*C1/2 + \
C2/2)*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2)))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t),
x(t) + y(t) + 9), Eq(diff(y(t), t), 2 * x(t) + 5 * y(t) + 23))
sol = [Eq(x(t), C1*exp(t*(-sqrt(6) + 3)) + C2*exp(t*(sqrt(6) + 3)) - \
Rational(22, 3)), Eq(y(t), C1*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) + C2*(2 + \
sqrt(6))*exp(t*(sqrt(6) + 3)) - Rational(5, 3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t),
x(t) + y(t) + 81), Eq(diff(y(t), t), -2 * x(t) + y(t) + 23))
sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - Rational(58, 3)), \
Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - Rational(185, 3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t), 5 * t * x(t) + 2 * y(t)),
Eq(diff(y(t), t), 2 * x(t) + 5 * t * y(t)))
sol = [Eq(x(t), (C1*exp((Integral(2, t).doit())) + C2*exp(-(Integral(2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (C1*exp((Integral(2, t)).doit()) - \
C2*exp(-(Integral(2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t), 5 * t * x(t) + t**2 * y(t)),
Eq(diff(y(t), t), -t**2 * x(t) + 5 * t * y(t)))
sol = [Eq(x(t), (C1*cos((Integral(t**2, t)).doit()) + C2*sin((Integral(t**2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (-C1*sin((Integral(t**2, t)).doit()) + \
C2*cos((Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t), 5 * t * x(t) + t**2 * y(t)),
Eq(diff(y(t), t), -t**2 * x(t) + (5 * t + 9 * t**2) * y(t)))
sol = [Eq(x(t), (C1*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
C2*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit())), \
Eq(y(t), (C1*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
C2*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t, t),
5 * x(t) + 43 * y(t)), Eq(diff(y(t), t, t),
x(t) + 9 * y(t)))
root0 = -sqrt(-sqrt(47) + 7)
root1 = sqrt(-sqrt(47) + 7)
root2 = -sqrt(sqrt(47) + 7)
root3 = sqrt(sqrt(47) + 7)
sol = [Eq(x(t), 43*C1*exp(t*root0) + 43*C2*exp(t*root1) + 43*C3*exp(t*root2) + 43*C4*exp(t*root3)), \
Eq(y(t), C1*(root0**2 - 5)*exp(t*root0) + C2*(root1**2 - 5)*exp(t*root1) + \
C3*(root2**2 - 5)*exp(t*root2) + C4*(root3**2 - 5)*exp(t*root3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t, t), 8 * x(t) + 3 * y(t) + 31),
Eq(diff(y(t), t, t), 9 * x(t) + 7 * y(t) + 12))
root0 = -sqrt(-sqrt(109) / 2 + Rational(15, 2))
root1 = sqrt(-sqrt(109) / 2 + Rational(15, 2))
root2 = -sqrt(sqrt(109) / 2 + Rational(15, 2))
root3 = sqrt(sqrt(109) / 2 + Rational(15, 2))
sol = [Eq(x(t), 3*C1*exp(t*root0) + 3*C2*exp(t*root1) + 3*C3*exp(t*root2) + 3*C4*exp(t*root3) - Rational(181, 29)), \
Eq(y(t), C1*(root0**2 - 8)*exp(t*root0) + C2*(root1**2 - 8)*exp(t*root1) + \
C3*(root2**2 - 8)*exp(t*root2) + C4*(root3**2 - 8)*exp(t*root3) + Rational(183, 29))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t, t) - 9 * diff(y(t), t) + 7 * x(t),
0), Eq(diff(y(t), t, t) + 9 * diff(x(t), t) + 7 * y(t), 0))
sol = [Eq(x(t), C1*cos(t*(Rational(9, 2) + sqrt(109)/2)) + C2*sin(t*(Rational(9, 2) + sqrt(109)/2)) + \
C3*cos(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*sin(t*(-sqrt(109)/2 + Rational(9, 2)))), Eq(y(t), -C1*sin(t*(Rational(9, 2) + sqrt(109)/2)) \
+ C2*cos(t*(Rational(9, 2) + sqrt(109)/2)) - C3*sin(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*cos(t*(-sqrt(109)/2 + Rational(9, 2))))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t, t), 9 * t * diff(y(t), t) - 9 * y(t)),
Eq(diff(y(t), t, t), 7 * t * diff(x(t), t) - 7 * x(t)))
I1 = sqrt(6) * 7**Rational(1, 4) * sqrt(pi) * erfi(
sqrt(6) * 7**Rational(1, 4) * t / 2) / 2 - exp(
3 * sqrt(7) * t**2 / 2) / t
I2 = -sqrt(6) * 7**Rational(1, 4) * sqrt(pi) * erf(
sqrt(6) * 7**Rational(1, 4) * t / 2) / 2 - exp(
-3 * sqrt(7) * t**2 / 2) / t
sol = [
Eq(x(t), C3 * t + t * (9 * C1 * I1 + 9 * C2 * I2)),
Eq(y(t), C4 * t + t * (3 * sqrt(7) * C1 * I1 - 3 * sqrt(7) * C2 * I2))
]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t), 21 * x(t)), Eq(diff(y(t), t),
17 * x(t) + 3 * y(t)),
Eq(diff(z(t), t), 5 * x(t) + 7 * y(t) + 9 * z(t)))
sol = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t), t),
3 * y(t) - 11 * z(t)), Eq(diff(y(t), t), 7 * z(t) - 3 * x(t)),
Eq(diff(z(t), t), 11 * x(t) - 7 * y(t)))
sol = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(3 * diff(x(t), t), 4 * 5 * (y(t) - z(t))),
Eq(4 * diff(y(t), t), 3 * 5 * (z(t) - x(t))),
Eq(5 * diff(z(t), t), 3 * 4 * (x(t) - y(t))))
sol = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t), t),
4 * x(t) - z(t)), Eq(diff(y(t), t), 2 * x(t) + 2 * y(t) - z(t)),
Eq(diff(z(t), t), 3 * x(t) + y(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t), t), 4 * x(t) - y(t) - 2 * z(t)),
Eq(diff(y(t), t),
2 * x(t) + y(t) - 2 * z(t)), Eq(diff(z(t), t),
5 * x(t) - 3 * z(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), \
Eq(y(t), C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), Eq(z(t), C1*exp(2*t) + 5*C2*cos(t) + 5*C3*sin(t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t), t), x(t) * y(t)**3), Eq(diff(y(t), t), y(t)**5))
sol = [Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t), t),
exp(3 * x(t)) * y(t)**3), Eq(diff(y(t), t),
y(t)**5))
sol = [Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), \
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(x(t),
t * diff(x(t), t) + diff(x(t), t) * diff(y(t), t)),
Eq(y(t),
t * diff(y(t), t) + diff(y(t), t)**2))
sol = set([Eq(x(t), C1 * C2 + C1 * t), Eq(y(t), C2**2 + C2 * t)])
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_erf():
x = Symbol("x")
e1 = sympy.erf(sympy.Symbol("x"))
e2 = erf(x)
assert sympify(e1) == e2
assert e2._sympy_() == e1
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)
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_moment_generating_function():
t = symbols('t', positive=True)
# Symbolic tests
a, b, c = symbols('a b c')
mgf = moment_generating_function(Beta('x', a, b))(t)
assert mgf == hyper((a, ), (a + b, ), t)
mgf = moment_generating_function(Chi('x', a))(t)
assert mgf == sqrt(2)*t*gamma(a/2 + S.Half)*\
hyper((a/2 + S.Half,), (Rational(3, 2),), t**2/2)/gamma(a/2) +\
hyper((a/2,), (S.Half,), t**2/2)
mgf = moment_generating_function(ChiSquared('x', a))(t)
assert mgf == (1 - 2 * t)**(-a / 2)
mgf = moment_generating_function(Erlang('x', a, b))(t)
assert mgf == (1 - t / b)**(-a)
mgf = moment_generating_function(ExGaussian("x", a, b, c))(t)
assert mgf == exp(a * t + b**2 * t**2 / 2) / (1 - t / c)
mgf = moment_generating_function(Exponential('x', a))(t)
assert mgf == a / (a - t)
mgf = moment_generating_function(Gamma('x', a, b))(t)
assert mgf == (-b * t + 1)**(-a)
mgf = moment_generating_function(Gumbel('x', a, b))(t)
assert mgf == exp(b * t) * gamma(-a * t + 1)
mgf = moment_generating_function(Gompertz('x', a, b))(t)
assert mgf == b * exp(b) * expint(t / a, b)
mgf = moment_generating_function(Laplace('x', a, b))(t)
assert mgf == exp(a * t) / (-b**2 * t**2 + 1)
mgf = moment_generating_function(Logistic('x', a, b))(t)
assert mgf == exp(a * t) * beta(-b * t + 1, b * t + 1)
mgf = moment_generating_function(Normal('x', a, b))(t)
assert mgf == exp(a * t + b**2 * t**2 / 2)
mgf = moment_generating_function(Pareto('x', a, b))(t)
assert mgf == b * (-a * t)**b * uppergamma(-b, -a * t)
mgf = moment_generating_function(QuadraticU('x', a, b))(t)
assert str(mgf) == (
"(3*(t*(-4*b + (a + b)**2) + 4)*exp(b*t) - "
"3*(t*(a**2 + 2*a*(b - 2) + b**2) + 4)*exp(a*t))/(t**2*(a - b)**3)")
mgf = moment_generating_function(RaisedCosine('x', a, b))(t)
assert mgf == pi**2 * exp(a * t) * sinh(b * t) / (b * t *
(b**2 * t**2 + pi**2))
mgf = moment_generating_function(Rayleigh('x', a))(t)
assert mgf == sqrt(2)*sqrt(pi)*a*t*(erf(sqrt(2)*a*t/2) + 1)\
*exp(a**2*t**2/2)/2 + 1
mgf = moment_generating_function(Triangular('x', a, b, c))(t)
assert str(mgf) == ("(-2*(-a + b)*exp(c*t) + 2*(-a + c)*exp(b*t) + "
"2*(b - c)*exp(a*t))/(t**2*(-a + b)*(-a + c)*(b - c))")
mgf = moment_generating_function(Uniform('x', a, b))(t)
assert mgf == (-exp(a * t) + exp(b * t)) / (t * (-a + b))
mgf = moment_generating_function(UniformSum('x', a))(t)
assert mgf == ((exp(t) - 1) / t)**a
mgf = moment_generating_function(WignerSemicircle('x', a))(t)
assert mgf == 2 * besseli(1, a * t) / (a * t)
# Numeric tests
mgf = moment_generating_function(Beta('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == hyper((2, ), (3, ), 1) / 2
mgf = moment_generating_function(Chi('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == sqrt(2) * hyper(
(1, ), (Rational(3, 2), ), S.Half) / sqrt(pi) + hyper(
(Rational(3, 2), ),
(Rational(3, 2), ), S.Half) + 2 * sqrt(2) * hyper(
(2, ), (Rational(5, 2), ), S.Half) / (3 * sqrt(pi))
mgf = moment_generating_function(ChiSquared('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == I
mgf = moment_generating_function(Erlang('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(ExGaussian("x", 0, 1, 1))(t)
assert mgf.diff(t).subs(t, 2) == -exp(2)
mgf = moment_generating_function(Exponential('x', 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Gamma('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Gumbel('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == EulerGamma + 1
mgf = moment_generating_function(Gompertz('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == -e * meijerg(((), (1, 1)),
((0, 0, 0), ()), 1)
mgf = moment_generating_function(Laplace('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Logistic('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == beta(1, 1)
mgf = moment_generating_function(Normal('x', 0, 1))(t)
assert mgf.diff(t).subs(t, 1) == exp(S.Half)
mgf = moment_generating_function(Pareto('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == expint(1, 0)
mgf = moment_generating_function(QuadraticU('x', 1, 2))(t)
assert mgf.diff(t).subs(t, 1) == -12 * e - 3 * exp(2)
mgf = moment_generating_function(RaisedCosine('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == -2*e*pi**2*sinh(1)/\
(1 + pi**2)**2 + e*pi**2*cosh(1)/(1 + pi**2)
mgf = moment_generating_function(Rayleigh('x', 1))(t)
assert mgf.diff(t).subs(t, 0) == sqrt(2) * sqrt(pi) / 2
mgf = moment_generating_function(Triangular('x', 1, 3, 2))(t)
assert mgf.diff(t).subs(t, 1) == -e + exp(3)
mgf = moment_generating_function(Uniform('x', 0, 1))(t)
assert mgf.diff(t).subs(t, 1) == 1
mgf = moment_generating_function(UniformSum('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == 1
mgf = moment_generating_function(WignerSemicircle('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == -2*besseli(1, 1) + besseli(2, 1) +\
besseli(0, 1)
def test_union():
N = Normal('N', 3, 2)
assert simplify(P(N**2 - N > 2)) == \
-erf(sqrt(2))/2 - erfc(sqrt(2)/4)/2 + 3/2
assert simplify(P(N**2 - 4 > 0)) == \
-erf(5*sqrt(2)/4)/2 - erfc(sqrt(2)/4)/2 + 3/2
def test_Or():
N = Normal('N', 0, 1)
assert simplify(P(Or(N > 2, N < 1))) == \
-erf(sqrt(2))/2 - erfc(sqrt(2)/2)/2 + 3/2
assert P(Or(N < 0, N < 1)) == P(N < 1)
assert P(Or(N > 0, N < 0)) == 1
def test_expr_to_holonomic():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic((sin(x) / x)**2)
q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \
[1, 0, -2/3])
assert p == q
p = expr_to_holonomic(1 / (1 + x**2)**2)
q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, [1])
assert p == q
p = expr_to_holonomic(exp(x) * sin(x) + x * log(1 + x))
q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \
- 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \
(-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \
7*x**2/2 + x + 5/2)*Dx**4, x, 0, [0, 1, 4, -1])
assert p == q
p = expr_to_holonomic(x * exp(x) + cos(x) + 1)
q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \
0, [2, 1, 1, 3])
assert p == q
assert (x * exp(x) + cos(x) + 1).series(n=10) == p.series(n=10)
p = expr_to_holonomic(log(1 + x)**2 + 1)
q = HolonomicFunction(
Dx + (3 * x + 3) * Dx**2 + (x**2 + 2 * x + 1) * Dx**3, x, 0, [1, 0, 2])
assert p == q
p = expr_to_holonomic(erf(x)**2 + x)
q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \
(x**2+ 1/4)*Dx**4, x, 0, [0, 1, 8/pi, 0])
assert p == q
p = expr_to_holonomic(cosh(x) * x)
q = HolonomicFunction((-x**2 + 2) - 2 * x * Dx + x**2 * Dx**2, x, 0,
[0, 1])
assert p == q
p = expr_to_holonomic(besselj(2, x))
q = HolonomicFunction((x**2 - 4) + x * Dx + x**2 * Dx**2, x, 0, [0, 0])
assert p == q
p = expr_to_holonomic(besselj(0, x) + exp(x))
q = HolonomicFunction((-x**2 - x/2 + 1/2) + (x**2 - x/2 - 3/2)*Dx + (-x**2 + x/2 + 1)*Dx**2 +\
(x**2 + x/2)*Dx**3, x, 0, [2, 1, 1/2])
assert p == q
p = expr_to_holonomic(sin(x)**2 / x)
q = HolonomicFunction(4 + 4 * x * Dx + 3 * Dx**2 + x * Dx**3, x, 0,
[0, 1, 0])
assert p == q
p = expr_to_holonomic(sin(x)**2 / x, x0=2)
q = HolonomicFunction((4) + (4 * x) * Dx + (3) * Dx**2 + (x) * Dx**3, x, 2,
[
sin(2)**2 / 2,
sin(2) * cos(2) - sin(2)**2 / 4,
-3 * sin(2)**2 / 4 + cos(2)**2 - sin(2) * cos(2)
])
assert p == q
p = expr_to_holonomic(log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2)
q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \
[-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0])
assert p == q
p = p.to_expr()
q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2
assert p == q
p = expr_to_holonomic(x**(S(1) / 2), x0=1)
q = HolonomicFunction(x * Dx - 1 / 2, x, 1, [1])
assert p == q
p = expr_to_holonomic(sqrt(1 + x**2))
q = HolonomicFunction((-x) + (x**2 + 1) * Dx, x, 0, [1])
assert p == q
assert (expr_to_holonomic(sqrt(x) + sqrt(2*x)).to_expr()-\
(sqrt(x) + sqrt(2*x))).simplify() == 0
assert expr_to_holonomic(3 * x +
2 * sqrt(x)).to_expr() == 3 * x + 2 * sqrt(x)
p = expr_to_holonomic((x**4 + x**3 + 5 * x**2 + 3 * x + 2) / x**2,
lenics=3)
q = HolonomicFunction((-2*x**4 - x**3 + 3*x + 4) + (x**5 + x**4 + 5*x**3 + 3*x**2 + \
2*x)*Dx, x, 0, {-2: [2, 3, 5]})
assert p == q
p = expr_to_holonomic(1 / (x - 1)**2, lenics=3, x0=1)
q = HolonomicFunction((2) + (x - 1) * Dx, x, 1, {-2: [1, 0, 0]})
assert p == q
a = symbols("a")
p = expr_to_holonomic(sqrt(a * x), x=x)
assert p.to_expr() == sqrt(a) * sqrt(x)
def test_erf_series():
assert erf(x).series(x, 0, 7) == 2*x/sqrt(pi) - \
2*x**3/3/sqrt(pi) + x**5/5/sqrt(pi) + O(x**7)
def test_mellin_transform():
from sympy import Max, Min, Ne
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 + S(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 + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)
def test_fresnel():
assert fresnels(0) == 0
assert fresnels(oo) == S.Half
assert fresnels(-oo) == -S.Half
assert fresnels(z) == fresnels(z)
assert fresnels(-z) == -fresnels(z)
assert fresnels(I * z) == -I * fresnels(z)
assert fresnels(-I * z) == I * fresnels(z)
assert conjugate(fresnels(z)) == fresnels(conjugate(z))
assert fresnels(z).diff(z) == sin(pi * z**2 / 2)
assert fresnels(z).rewrite(erf) == (S.One + I) / 4 * (erf(
(S.One + I) / 2 * sqrt(pi) * z) - I * erf(
(S.One - I) / 2 * sqrt(pi) * z))
assert fresnels(z).rewrite(hyper) == \
pi*z**3/6 * hyper([S(3)/4], [S(3)/2, S(7)/4], -pi**2*z**4/16)
assert fresnels(z).series(z, n=15) == \
pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15)
assert fresnels(w).is_real is True
assert fresnels(z).as_real_imag() == \
((fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 +
fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2,
I*(fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) -
fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) *
re(z)*Abs(im(z))/(2*im(z)*Abs(re(z)))))
assert fresnels(2 + 3 * I).as_real_imag() == (
fresnels(2 + 3 * I) / 2 + fresnels(2 - 3 * I) / 2,
I * (fresnels(2 - 3 * I) - fresnels(2 + 3 * I)) / 2)
assert expand_func(integrate(fresnels(z), z)) == \
z*fresnels(z) + cos(pi*z**2/2)/pi
assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(9)/4) * \
meijerg(((), (1,)), ((S(3)/4,),
(S(1)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(3)/4)*(z**2)**(S(3)/4))
assert fresnelc(0) == 0
assert fresnelc(oo) == S.Half
assert fresnelc(-oo) == -S.Half
assert fresnelc(z) == fresnelc(z)
assert fresnelc(-z) == -fresnelc(z)
assert fresnelc(I * z) == I * fresnelc(z)
assert fresnelc(-I * z) == -I * fresnelc(z)
assert conjugate(fresnelc(z)) == fresnelc(conjugate(z))
assert fresnelc(z).diff(z) == cos(pi * z**2 / 2)
assert fresnelc(z).rewrite(erf) == (S.One - I) / 4 * (erf(
(S.One + I) / 2 * sqrt(pi) * z) + I * erf(
(S.One - I) / 2 * sqrt(pi) * z))
assert fresnelc(z).rewrite(hyper) == \
z * hyper([S.One/4], [S.One/2, S(5)/4], -pi**2*z**4/16)
assert fresnelc(z).series(z, n=15) == \
z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15)
assert fresnelc(w).is_real is True
assert fresnelc(z).as_real_imag() == \
((fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 +
fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2,
I*(fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) -
fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) *
re(z)*Abs(im(z))/(2*im(z)*Abs(re(z)))))
assert fresnelc(2 + 3 * I).as_real_imag() == (
fresnelc(2 - 3 * I) / 2 + fresnelc(2 + 3 * I) / 2,
I * (fresnelc(2 - 3 * I) - fresnelc(2 + 3 * I)) / 2)
assert expand_func(integrate(fresnelc(z), z)) == \
z*fresnelc(z) - sin(pi*z**2/2)/pi
assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(3)/4) * \
meijerg(((), (1,)), ((S(1)/4,),
(S(3)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(1)/4)*(z**2)**(S(1)/4))
from sympy.utilities.randtest import test_numerically
test_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z)
test_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z)
test_numerically(fresnels(z), fresnels(z).rewrite(hyper), z)
test_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z)
test_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z)
test_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z)
test_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z)
test_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
def test_laplace_transform():
from sympy import (fresnels, fresnelc, hyper)
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) + log(s) + EulerGamma) / (-s), 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) == (1 / b / (1 + (a + s)**2 / b**2), -a, True)
assert LT(exp(-a*t)*cos(b*t), t, s) == \
(1/(s + a)/(1 + b**2/(a + s)**2), -a, True)
# TODO sinh, cosh have delicate cancellation
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) - sin(s**2/(2*pi))/2 -
cos(s**2/(2*pi))*fresnels(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True)
'cos': lambda x: sympy.cos(x),
'sin': lambda x: sympy.sin(x),
'tan': lambda x: sympy.tan(x),
'cosh': lambda x: sympy.cosh(x),
'sinh': lambda x: sympy.sinh(x),
'tanh': lambda x: sympy.tanh(x),
'exp': lambda x: sympy.exp(x),
'acos': lambda x: sympy.acos(x),
'asin': lambda x: sympy.asin(x),
'atan': lambda x: sympy.atan(x),
'acosh': lambda x: sympy.acosh(x),
'asinh': lambda x: sympy.asinh(x),
'atanh': lambda x: sympy.atanh(x),
'abs': lambda x: abs(x),
'mod': lambda x, y: sympy.Mod(x, y),
'erf': lambda x: sympy.erf(x),
'erfc': lambda x: sympy.erfc(x),
'logm': lambda x: sympy.log(abs(x)),
'logm10': lambda x: sympy.log10(abs(x)),
'logm2': lambda x: sympy.log2(abs(x)),
'log1p': lambda x: sympy.log(x + 1),
'floor': lambda x: sympy.floor(x),
'ceil': lambda x: sympy.ceil(x),
'sign': lambda x: sympy.sign(x),
'round': lambda x: sympy.round(x),
}
def pysr(
X=None,
y=None,
def test_issue_1791():
z = Symbol('z', positive=True)
assert integrate(exp(-log(x)**2),x) == pi**(S(1)/2)*erf(-S(1)/2 + log(x))*exp(S(1)/4)/2
assert integrate(exp(log(x)**2),x) == -I*pi**(S(1)/2)*erf(I*log(x) + I/2)*exp(-S(1)/4)/2
assert integrate(exp(-z*log(x)**2),x) == \
pi**(S(1)/2)*erf(z**(S(1)/2)*log(x) - 1/(2*z**(S(1)/2)))*exp(S(1)/(4*z))/(2*z**(S(1)/2))
def test_erf2():
assert erf2(0, 0) == S.Zero
assert erf2(x, x) == S.Zero
assert erf2(nan, 0) == nan
assert erf2(-oo, y) == erf(y) + 1
assert erf2(oo, y) == erf(y) - 1
assert erf2(x, oo) == 1 - erf(x)
assert erf2(x, -oo) == -1 - erf(x)
assert erf2(x, erf2inv(x, y)) == y
assert erf2(-x, -y) == -erf2(x, y)
assert erf2(-x, y) == erf(y) + erf(x)
assert erf2(x, -y) == -erf(y) - erf(x)
assert erf2(x, y).rewrite(
'fresnels') == erf(y).rewrite(fresnels) - erf(x).rewrite(fresnels)
assert erf2(x, y).rewrite(
'fresnelc') == erf(y).rewrite(fresnelc) - erf(x).rewrite(fresnelc)
assert erf2(
x, y).rewrite('hyper') == erf(y).rewrite(hyper) - erf(x).rewrite(hyper)
assert erf2(x, y).rewrite(
'meijerg') == erf(y).rewrite(meijerg) - erf(x).rewrite(meijerg)
assert erf2(
x,
y).rewrite('uppergamma'
) == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
assert erf2(
x,
y).rewrite('expint') == erf(y).rewrite(expint) - erf(x).rewrite(expint)
assert erf2(I, 0).is_real is False
assert erf2(0, 0).is_real is True
#assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))
assert erf2(x, y).rewrite('erf') == erf(y) - erf(x)
assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
assert erf2(x, y).rewrite('erfi') == I * (erfi(I * x) - erfi(I * y))
raises(ArgumentIndexError, lambda: erfi(x).fdiff(3))
def test_fresnel():
assert fresnels(0) == 0
assert fresnels(oo) == S.Half
assert fresnels(-oo) == Rational(-1, 2)
assert fresnels(I * oo) == -I * S.Half
assert unchanged(fresnels, z)
assert fresnels(-z) == -fresnels(z)
assert fresnels(I * z) == -I * fresnels(z)
assert fresnels(-I * z) == I * fresnels(z)
assert conjugate(fresnels(z)) == fresnels(conjugate(z))
assert fresnels(z).diff(z) == sin(pi * z**2 / 2)
assert fresnels(z).rewrite(erf) == (S.One + I) / 4 * (erf(
(S.One + I) / 2 * sqrt(pi) * z) - I * erf(
(S.One - I) / 2 * sqrt(pi) * z))
assert fresnels(z).rewrite(hyper) == \
pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16)
assert fresnels(z).series(z, n=15) == \
pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15)
assert fresnels(w).is_extended_real is True
assert fresnels(w).is_finite is True
assert fresnels(z).is_extended_real is None
assert fresnels(z).is_finite is None
assert fresnels(z).as_real_imag() == (
fresnels(re(z) - I * im(z)) / 2 + fresnels(re(z) + I * im(z)) / 2,
-I * (-fresnels(re(z) - I * im(z)) + fresnels(re(z) + I * im(z))) / 2)
assert fresnels(z).as_real_imag(deep=False) == (
fresnels(re(z) - I * im(z)) / 2 + fresnels(re(z) + I * im(z)) / 2,
-I * (-fresnels(re(z) - I * im(z)) + fresnels(re(z) + I * im(z))) / 2)
assert fresnels(w).as_real_imag() == (fresnels(w), 0)
assert fresnels(w).as_real_imag(deep=True) == (fresnels(w), 0)
assert fresnels(2 + 3 * I).as_real_imag() == (
fresnels(2 + 3 * I) / 2 + fresnels(2 - 3 * I) / 2,
-I * (fresnels(2 + 3 * I) - fresnels(2 - 3 * I)) / 2)
assert expand_func(integrate(fresnels(z), z)) == \
z*fresnels(z) + cos(pi*z**2/2)/pi
assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(9, 4) * \
meijerg(((), (1,)), ((Rational(3, 4),),
(Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4))
assert fresnelc(0) == 0
assert fresnelc(oo) == S.Half
assert fresnelc(-oo) == Rational(-1, 2)
assert fresnelc(I * oo) == I * S.Half
assert unchanged(fresnelc, z)
assert fresnelc(-z) == -fresnelc(z)
assert fresnelc(I * z) == I * fresnelc(z)
assert fresnelc(-I * z) == -I * fresnelc(z)
assert conjugate(fresnelc(z)) == fresnelc(conjugate(z))
assert fresnelc(z).diff(z) == cos(pi * z**2 / 2)
assert fresnelc(z).rewrite(erf) == (S.One - I) / 4 * (erf(
(S.One + I) / 2 * sqrt(pi) * z) + I * erf(
(S.One - I) / 2 * sqrt(pi) * z))
assert fresnelc(z).rewrite(hyper) == \
z * hyper([Rational(1, 4)], [S.Half, Rational(5, 4)], -pi**2*z**4/16)
assert fresnelc(w).is_extended_real is True
assert fresnelc(z).as_real_imag() == \
(fresnelc(re(z) - I*im(z))/2 + fresnelc(re(z) + I*im(z))/2,
-I*(-fresnelc(re(z) - I*im(z)) + fresnelc(re(z) + I*im(z)))/2)
assert fresnelc(z).as_real_imag(deep=False) == \
(fresnelc(re(z) - I*im(z))/2 + fresnelc(re(z) + I*im(z))/2,
-I*(-fresnelc(re(z) - I*im(z)) + fresnelc(re(z) + I*im(z)))/2)
assert fresnelc(2 + 3 * I).as_real_imag() == (
fresnelc(2 - 3 * I) / 2 + fresnelc(2 + 3 * I) / 2,
-I * (fresnelc(2 + 3 * I) - fresnelc(2 - 3 * I)) / 2)
assert expand_func(integrate(fresnelc(z), z)) == \
z*fresnelc(z) - sin(pi*z**2/2)/pi
assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(3, 4) * \
meijerg(((), (1,)), ((Rational(1, 4),),
(Rational(3, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(1, 4)*(z**2)**Rational(1, 4))
from sympy.testing.randtest import verify_numerically
verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z)
verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z)
verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z)
verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z)
verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z)
verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z)
verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z)
verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
raises(ArgumentIndexError, lambda: fresnels(z).fdiff(2))
raises(ArgumentIndexError, lambda: fresnelc(z).fdiff(2))
assert fresnels(x).taylor_term(-1, x) is S.Zero
assert fresnelc(x).taylor_term(-1, x) is S.Zero
assert fresnelc(x).taylor_term(1, x) == -pi**2 * x**5 / 40
def test_issue_10976():
s, x = symbols('s x', real=True)
assert limit(erf(s*x)/erf(s), s, 0) == x
def test_erf():
assert erf(nan) is nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) == 0
assert erf(I * oo) == oo * I
assert erf(-I * oo) == -oo * I
assert erf(-2) == -erf(2)
assert erf(-x * y) == -erf(x * y)
assert erf(-x - y) == -erf(x + y)
assert erf(erfinv(x)) == x
assert erf(erfcinv(x)) == 1 - x
assert erf(erf2inv(0, x)) == x
assert erf(erf2inv(0, x, evaluate=False)) == x # To cover code in erf
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_real is False
assert erf(0).is_real is True
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi)
assert erf(x * y).as_leading_term(y) == 2 * x * y / sqrt(pi)
assert (erf(x * y) / erf(y)).as_leading_term(y) == x
assert erf(1 / x).as_leading_term(x) == erf(1 / x)
assert erf(z).rewrite('uppergamma') == sqrt(z**
2) * (1 - erfc(sqrt(z**2))) / z
assert erf(z).rewrite('erfc') == S.One - erfc(z)
assert erf(z).rewrite('erfi') == -I * erfi(I * z)
assert erf(z).rewrite('fresnels') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('fresnelc') == (1 + I) * (
fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z *
(1 - I) / sqrt(pi)))
assert erf(z).rewrite('hyper') == 2 * z * hyper([S.Half], [3 * S.Half],
-z**2) / sqrt(pi)
assert erf(z).rewrite('meijerg') == z * meijerg(
[S.Half], [], [0], [Rational(-1, 2)], z**2) / sqrt(pi)
assert erf(z).rewrite(
'expint') == sqrt(z**2) / z - z * expint(S.Half, z**2) / sqrt(S.Pi)
assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
2/sqrt(pi)
assert limit((1 - erf(z)) * exp(z**2) * z, z, oo) == 1 / sqrt(pi)
assert limit((1 - erf(x)) * exp(x**2) * sqrt(pi) * x, x, oo) == 1
assert limit(((1 - erf(x)) * exp(x**2) * sqrt(pi) * x - 1) * 2 * x**2, x,
oo) == -1
assert limit(erf(x) / x, x, 0) == 2 / sqrt(pi)
assert limit(x**(-4) - sqrt(pi) * erf(x**2) / (2 * x**6), x, 0) == S(1) / 3
assert erf(x).as_real_imag() == \
(erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
-I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
assert erf(x).as_real_imag(deep=False) == \
(erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2,
-I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2)
assert erf(w).as_real_imag() == (erf(w), 0)
assert erf(w).as_real_imag(deep=False) == (erf(w), 0)
# issue 13575
assert erf(I).as_real_imag() == (0, -I * erf(I))
raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
assert erf(x).inverse() == erfinv
def test_inverse_mellin_transform():
from sympy import (sin, simplify, expand_func, powsimp, Max, Min, expand,
powdenest, powsimp, exp_polar, combsimp, 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*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
# 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', 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)
c, d = symbols('c d')
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)
from sympy import expand_mul
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
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 + sqrt(
b**2 + x))**(a - 1)*(b**2 + b*sqrt(b**2 + x) + 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 + sqrt(b**2 + x))**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,
(-S(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 sympy 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 + S(1)/2)/(sqrt(pi)*s), s, x, (-S(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, S(3)/4))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(S(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, S(1)/4))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
/ (gamma(S(1)/2 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, S(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(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), S(1)/2))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, S(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, S(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), S(1)/2))) == \
(-cos(pi*b)*besselj(b, sqrt(x)) + besselj(-b, sqrt(x))) * \
besselj(a, sqrt(x))/sin(pi*b)*(-1)
# TODO more
# for coverage
assert IMT(pi / cos(pi * s), s, x, (0, S(1) / 2)) == sqrt(x) / (x + 1)
def test_erf_evalf():
assert abs(erf(Float(2.0)) - 0.995322265) < 1E-8 # XXX
if self._lazy:
return value
_simplify = getattr(value, self._SIMPLIFY, None)
if _simplify is not None:
try:
return _simplify(**kwargs)
finally:
print('Formula simplified.')
def __ror__(self, value):
return self.__call__(value)
simplify = Simplify(lazy=False)
# In[3]:
S, E, r, σ, T, t = symbols('S, E, r, σ, T, t')
_ = symbols('_')
d1 = (log(S / E) + r * (T - t) + σ**2 * (T - t) / 2) / (σ * sqrt(T - t))
d2 = d1 - σ * sqrt(T - t)
N = lambda x: (erf(sqrt(2) * x / 2) + 1) / 2
C = S * N(d1) - E * exp(-r * (T - t) * N(d2)) | simplify
Δ = diff(C, S)
ν = diff(C, σ)
θ = diff(C, t)
γ = diff(C, S, 2)
def test_mellin_transform():
from sympy 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, e.g.
# And(re(rho) - 1 < 0, re(rho) < 1) should just be
# re(rho) < 1
assert MT(abs(1 - x)**(-rho), x, s) == (
2*sin(pi*rho/2)*gamma(1 - rho)*
cos(pi*(rho/2 - s))*gamma(s)*gamma(rho-s)/pi,
(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), re(beta) > 0)
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 + S(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)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)
def test_erf2():
assert erf2(0, 0) is S.Zero
assert erf2(x, x) is S.Zero
assert erf2(nan, 0) is nan
assert erf2(-oo, y) == erf(y) + 1
assert erf2(oo, y) == erf(y) - 1
assert erf2(x, oo) == 1 - erf(x)
assert erf2(x, -oo) == -1 - erf(x)
assert erf2(x, erf2inv(x, y)) == y
assert erf2(-x, -y) == -erf2(x, y)
assert erf2(-x, y) == erf(y) + erf(x)
assert erf2(x, -y) == -erf(y) - erf(x)
assert erf2(x, y).rewrite(
'fresnels') == erf(y).rewrite(fresnels) - erf(x).rewrite(fresnels)
assert erf2(x, y).rewrite(
'fresnelc') == erf(y).rewrite(fresnelc) - erf(x).rewrite(fresnelc)
assert erf2(
x, y).rewrite('hyper') == erf(y).rewrite(hyper) - erf(x).rewrite(hyper)
assert erf2(x, y).rewrite(
'meijerg') == erf(y).rewrite(meijerg) - erf(x).rewrite(meijerg)
assert erf2(
x,
y).rewrite('uppergamma'
) == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
assert erf2(
x,
y).rewrite('expint') == erf(y).rewrite(expint) - erf(x).rewrite(expint)
assert erf2(I, 0).is_real is False
assert erf2(0, 0).is_real is True
assert expand_func(erf(x) + erf2(x, y)) == erf(y)
assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))
assert erf2(x, y).rewrite('erf') == erf(y) - erf(x)
assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
assert erf2(x, y).rewrite('erfi') == I * (erfi(I * x) - erfi(I * y))
assert erf2(x, y).diff(x) == erf2(x, y).fdiff(1)
assert erf2(x, y).diff(y) == erf2(x, y).fdiff(2)
assert erf2(x, y).diff(x) == -2 * exp(-x**2) / sqrt(pi)
assert erf2(x, y).diff(y) == 2 * exp(-y**2) / sqrt(pi)
raises(ArgumentIndexError, lambda: erf2(x, y).fdiff(3))
assert erf2(x, y).is_extended_real is None
xr, yr = symbols('xr yr', extended_real=True)
assert erf2(xr, yr).is_extended_real is True
def test_pmint_erf():
f = exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1)
g = sqrt(pi) * log(erf(x) - 1) / 8 - sqrt(pi) * log(erf(x) + 1) / 8 - sqrt(
pi) / (4 * erf(x) - 4)
assert ratsimp(heurisch(f, x)) == g
def test_issue_841():
a, b, c, d = symbols('a:d', positive=True, bounded=True)
assert integrate(exp(-x**2 + I*c*x), x) == \
-sqrt(pi)*exp(-c**2/4)*erf(I*c/2 - x)/2
assert integrate(exp(a*x**2 + b*x + c), x) == \
sqrt(pi)*exp(c)*exp(-b**2/(4*a))*erfi(sqrt(a)*x + b/(2*sqrt(a)))/(2*sqrt(a))
def test_risch_option():
# risch=True only allowed on indefinite integrals
raises(ValueError, lambda: integrate(1/log(x), (x, 0, oo), risch=True))
assert integrate(exp(-x**2), x, risch=True) == NonElementaryIntegral(exp(-x**2), x)
assert integrate(log(1/x)*y, x, y, risch=True) == y**2*(x*log(1/x)/2 + x/2)
assert integrate(erf(x), x, risch=True) == Integral(erf(x), x)
