def test_to_sympy():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, 1).to_sympy()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_sympy()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_sympy()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_sympy()
q = 1 / (x**2 - 2 * x + 1)
assert p == q
p = from_sympy(sin(x)**2 / x).integrate((x, 0, x)).to_sympy()
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 = from_sympy(log(1 + x**2)).to_sympy()
q = C_2 * log(x**2 + 1)
assert p == q
p = from_sympy(log(1 + x**2)).diff().to_sympy()
q = C_1 * x / (x**2 + 1)
assert p == q
p = from_sympy(erf(x) + x).to_sympy()
q = 3 * C_3 * x - 3 * sqrt(pi) * C_3 * erf(x) / 2 + x + 2 * x / sqrt(pi)
assert p == q
def test_multiplication_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 + x*Dx - 1, x, 0, [3, 1])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \
(2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3])
assert p * q == r
p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3])
r = HolonomicFunction((x**8 - 37*x**7/27 - 10*x**6/27 - 164*x**5/9 - 184*x**4/9 + \
160*x**3/27 + 404*x**2/9 + 8*x + 40/3) + (6*x**7 - 128*x**6/9 - 98*x**5/9 - 28*x**4/9 + \
8*x**3/9 + 28*x**2 + 40*x/9 - 40)*Dx + (3*x**6 - 82*x**5/9 + 76*x**4/9 + 4*x**3/3 + \
220*x**2/9 - 80*x/3)*Dx**2 + (-2*x**6 + 128*x**5/27 - 2*x**4/3 -80*x**2/9 + 200/9)*Dx**3 + \
(3*x**5 - 64*x**4/9 - 28*x**3/9 + 6*x**2 - 20*x/9 - 20/3)*Dx**4 + (-4*x**3 + 64*x**2/9 + \
8*x/3)*Dx**5 + (x**4 - 64*x**3/27 - 4*x**2/3 + 20/9)*Dx**6, x, 0, [3, 3, 3, -3, -12, -24])
assert p * q == r
p = HolonomicFunction(Dx - 1, x, 0, [2])
q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
r = HolonomicFunction(2 -2*Dx + Dx**2, x, 0, [0, 2])
assert p * q == r
q = HolonomicFunction(x*Dx**2 + 1 + 2*Dx, x, 0,[0, 1])
r = HolonomicFunction((x - 1) + (-2*x + 2)*Dx + x*Dx**2, x, 0, [0, 2])
assert p * q == r
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3])
q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1])
r = HolonomicFunction(6*Dx + 3*Dx**2 + 2*Dx**3 - 3*Dx**4 + Dx**6, x, 0, [1, 5, 14, 17, 17, 2])
assert p * q == r
p = from_sympy(sin(x))
q = from_sympy(1/x)
r = HolonomicFunction(x + 2*Dx + x*Dx**2, x, 1, [sin(1), -sin(1) + cos(1)])
assert p * q == r
def test_to_sympy():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, 1).to_sympy()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_sympy()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_sympy()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_sympy()
q = 1/(x**2 - 2*x + 1)
assert p == q
p = from_sympy(sin(x)**2/x).integrate((x, 0, x)).to_sympy()
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 = from_sympy(log(1+x**2)).to_sympy()
q = C_2*log(x**2 + 1)
assert p == q
p = from_sympy(log(1+x**2)).diff().to_sympy()
q = C_1*x/(x**2 + 1)
assert p == q
p = from_sympy(erf(x) + x).to_sympy()
q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi)
assert p == q
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), 3)
assert p == q
C_1, C_2, C_3 = symbols('C_1, C_2, C_3')
p = from_sympy(log(1+x**2))
q = (HolonomicSequence(n**2 + (n**2 + 2*n)*Sn**2, [0, 0, C_2, 0]), 2)
assert p.to_sequence() == q
p = p.diff()
q = (HolonomicSequence((n + 1) + (n + 1)*Sn**2, [0, C_1, 0]), 1)
assert p.to_sequence() == q
p = from_sympy(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]), 2)
assert p == q
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), 3)
assert p == q
C_1, C_2, C_3 = symbols('C_1, C_2, C_3')
p = from_sympy(log(1 + x**2))
q = (HolonomicSequence(n**2 + (n**2 + 2 * n) * Sn**2, [0, 0, C_2, 0]), 2)
assert p.to_sequence() == q
p = p.diff()
q = (HolonomicSequence((n + 1) + (n + 1) * Sn**2, [0, C_1, 0]), 1)
assert p.to_sequence() == q
p = from_sympy(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]), 2)
assert p == q
def test_multiplication_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 + x * Dx - 1, x, 0, [3, 1])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \
(2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3])
assert p * q == r
p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3])
r = HolonomicFunction((x**8 - 37*x**7/27 - 10*x**6/27 - 164*x**5/9 - 184*x**4/9 + \
160*x**3/27 + 404*x**2/9 + 8*x + 40/3) + (6*x**7 - 128*x**6/9 - 98*x**5/9 - 28*x**4/9 + \
8*x**3/9 + 28*x**2 + 40*x/9 - 40)*Dx + (3*x**6 - 82*x**5/9 + 76*x**4/9 + 4*x**3/3 + \
220*x**2/9 - 80*x/3)*Dx**2 + (-2*x**6 + 128*x**5/27 - 2*x**4/3 -80*x**2/9 + 200/9)*Dx**3 + \
(3*x**5 - 64*x**4/9 - 28*x**3/9 + 6*x**2 - 20*x/9 - 20/3)*Dx**4 + (-4*x**3 + 64*x**2/9 + \
8*x/3)*Dx**5 + (x**4 - 64*x**3/27 - 4*x**2/3 + 20/9)*Dx**6, x, 0, [3, 3, 3, -3, -12, -24])
assert p * q == r
p = HolonomicFunction(Dx - 1, x, 0, [2])
q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
r = HolonomicFunction(2 - 2 * Dx + Dx**2, x, 0, [0, 2])
assert p * q == r
q = HolonomicFunction(x * Dx**2 + 1 + 2 * Dx, x, 0, [0, 1])
r = HolonomicFunction((x - 1) + (-2 * x + 2) * Dx + x * Dx**2, x, 0,
[0, 2])
assert p * q == r
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3])
q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1])
r = HolonomicFunction(6 * Dx + 3 * Dx**2 + 2 * Dx**3 - 3 * Dx**4 + Dx**6,
x, 0, [1, 5, 14, 17, 17, 2])
assert p * q == r
p = from_sympy(sin(x))
q = from_sympy(1 / x)
r = HolonomicFunction(x + 2 * Dx + x * Dx**2, x, 1,
[sin(1), -sin(1) + cos(1)])
assert p * q == r
def test_addition_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, 3)
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2])
assert p + q == r
p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2])
q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
r = HolonomicFunction((-x**4 - x**3/4 - x**2 + 1/4) + (x**3 + x**2/4 + 3*x/4 + 1)*Dx + \
(-3*x/2 + 7/4)*Dx**2 + (x**2 - 7*x/4 + 1/4)*Dx**3 + (x**2 + x/4 + 1/2)*Dx**4, x, 0, [2, 2, -2, 2])
assert p + q == r
p = HolonomicFunction(Dx**2 + 4 * x * Dx + x**2, x, 0, [3, 4])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \
(x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \
10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17])
assert p + q == r
q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1])
p = HolonomicFunction(Dx - 1, x, 2, [1])
r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \
(x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ])
assert p + q == r
p = from_sympy(sin(x))
q = from_sympy(1 / x)
r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \
x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2])
assert p + q == r
def test_addition_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx-1, x, 0, 3)
q = HolonomicFunction(Dx**2+1, x, 0, [1, 0])
r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2])
assert p + q == r
p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2])
q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
r = HolonomicFunction((-x**4 - x**3/4 - x**2 + 1/4) + (x**3 + x**2/4 + 3*x/4 + 1)*Dx + \
(-3*x/2 + 7/4)*Dx**2 + (x**2 - 7*x/4 + 1/4)*Dx**3 + (x**2 + x/4 + 1/2)*Dx**4, x, 0, [2, 2, -2, 2])
assert p + q == r
p = HolonomicFunction(Dx**2 + 4*x*Dx + x**2, x, 0, [3, 4])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \
(x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \
10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17])
assert p + q == r
q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1])
p = HolonomicFunction(Dx - 1, x, 2, [1])
r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \
(x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ])
assert p + q == r
p = from_sympy(sin(x))
q = from_sympy(1/x)
r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \
x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2])
assert p + q == r
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_from_sympy():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = from_sympy((sin(x)/x)**2)
q = HolonomicFunction(24*x + (24*x**2 + 12)*Dx + (4*x**3 + 24*x)*Dx**2 + \
10*x**2*Dx**3 + x**3*Dx**4, x, 1, [-cos(2)/2 + 1/2, -1 + cos(2) + sin(2), \
-4*sin(2) - cos(2) + 3, -12 + 14*sin(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
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(24*x + (24*x**2 + 12)*Dx + (4*x**3 + 24*x)*Dx**2 + \
10*x**2*Dx**3 + x**3*Dx**4, x, 1, [-cos(2)/2 + 1/2, -1 + cos(2) + sin(2), \
-4*sin(2) - cos(2) + 3, -12 + 14*sin(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
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 = from_sympy(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
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 = from_sympy(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
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = from_sympy(sin(x)**2/x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = from_sympy(sin(x)).integrate((x, 0, x)).to_sympy()
q = 1 - cos(x)
assert p == q
p = from_sympy(sin(x)).integrate((x, 0, 3))
q = '1.98999246812687'
assert sstr(p) == q
p = from_sympy(sin(x)/x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = from_sympy(sin(x)**2/x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = from_sympy(sin(x)**2 / x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = from_sympy(sin(x)).integrate((x, 0, x)).to_sympy()
q = 1 - cos(x)
assert p == q
p = from_sympy(sin(x)).integrate((x, 0, 3))
q = '1.98999246812687'
assert sstr(p) == q
p = from_sympy(sin(x) / x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = from_sympy(sin(x)**2 / x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
def test_diff():
x, y = symbols('x, y')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(x*Dx**2 + 1, x, 0, [0, 1])
assert p.diff().to_sympy() == p.to_sympy().diff().simplify()
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0])
assert p.diff(x, 2).to_sympy() == p.to_sympy()
p = from_sympy(Si(x))
assert p.diff().to_sympy() == sin(x)/x
assert p.diff(y) == 0
C_1, C_2, C_3 = symbols('C_1, C_2, C_3')
q = Si(x)
assert p.diff(x).to_sympy() == q.diff()
assert p.diff(x, 2).to_sympy().subs(C_1, -S(1)/3) == q.diff(x, 2).simplify()
assert p.diff(x, 3).series().subs(C_2, S(1)/10) == q.diff(x, 3).series()
def test_diff():
x, y = symbols('x, y')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(x * Dx**2 + 1, x, 0, [0, 1])
assert p.diff().to_sympy() == p.to_sympy().diff().simplify()
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0])
assert p.diff(x, 2).to_sympy() == p.to_sympy()
p = from_sympy(Si(x))
assert p.diff().to_sympy() == sin(x) / x
assert p.diff(y) == 0
C_1, C_2, C_3 = symbols('C_1, C_2, C_3')
q = Si(x)
assert p.diff(x).to_sympy() == q.diff()
assert p.diff(x, 2).to_sympy().subs(C_1,
-S(1) / 3) == q.diff(x, 2).simplify()
assert p.diff(x, 3).series().subs(C_2, S(1) / 10) == q.diff(x, 3).series()
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, 0, \
[1, 0, -2/3])
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((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 = 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((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 = 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((-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 = from_sympy(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 = from_sympy(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 = from_sympy(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_sympy()
q = log(x)/2 - Ci(2*x)/2 + Ci(2)/2
assert p == q
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, 0, \
[1, 0, -2/3])
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((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 = 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((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 = 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((-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 = from_sympy(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 = from_sympy(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 = from_sympy(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_sympy()
q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2
assert p == q