def _eval_as_leading_term(self, x):
from sympy import Order
n, z = [a.as_leading_term(x) for a in self.args]
o = Order(z, x)
if n == 0 and o.contains(1/x):
return o.getn() * log(x)
else:
return self.func(n, z)
def test_indexed_series():
A = IndexedBase("A")
i = symbols("i", integer=True)
assert sin(A[i]).series(
A[i]) == A[i] - A[i]**3 / 6 + A[i]**5 / 120 + Order(A[i]**6, A[i])
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left (x \right )}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left (x,y \right )}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left (x,y,z \right )}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left (x \right )}'
beta = Function('beta')
# not to be confused with the beta function
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(beta) == r"\beta"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(FallingFactorial(3,
k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma{\left(x \right)}"
w = Wild('w')
assert latex(gamma(w)) == r"\Gamma{\left(w \right)}"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(Order(x, x)) == r"\mathcal{O}\left(x\right)"
assert latex(Order(x, x, 0)) == r"\mathcal{O}\left(x\right)"
assert latex(Order(x, x,
oo)) == r"\mathcal{O}\left(x; x\rightarrow\infty\right)"
assert latex(
Order(x, x, y)
) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)"
assert latex(
Order(x, x, y, 0)
) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)"
assert latex(
Order(x, x, y, oo)
) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow\infty\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(polylog(x,
y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
assert latex(Chi(x)) == r'\operatorname{Chi}{\left (x \right )}'
assert latex(jacobi(n, a, b,
x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(
n, a, b,
x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a,
x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(
n, a,
x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n,
x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n,
x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a,
x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(
n, a,
x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a,
x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(
n, a,
x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(
Ynm(n, m, theta,
phi)**3) == r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(
Znm(n, m, theta,
phi)**3) == r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(
polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
assert latex(polar_lift(0)**
3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
assert latex(totient(n)) == r'\phi\left( n \right)'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left (x \right )}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
def test_simple_6():
assert Order(x) - Order(x) == Order(x)
assert Order(x) + Order(1) == Order(1)
assert Order(x) + Order(x**2) == Order(x)
assert Order(1/x) + Order(1) == Order(1/x)
assert Order(x) + Order(exp(1/x)) == Order(exp(1/x))
assert Order(x**3) + Order(exp(2/x)) == Order(exp(2/x))
assert Order(x**-3) + Order(exp(2/x)) == Order(exp(2/x))
def test_simple_4():
assert Order(x)**2 == Order(x**2)
def test_simple_2():
assert Order(2*x)*x == Order(x**2)
assert Order(2*x)/x == Order(1, x)
assert Order(2*x)*x*exp(1/x) == Order(x**2*exp(1/x))
assert (Order(2*x)*x*exp(1/x)/ln(x)**3).expr == x**2*exp(1/x)*ln(x)**-3
def test_order_at_some_point():
assert Order(x, (x, 1)) == Order(1, (x, 1))
assert Order(2*x - 2, (x, 1)) == Order(x - 1, (x, 1))
assert Order(-x + 1, (x, 1)) == Order(x - 1, (x, 1))
assert Order(x - 1, (x, 1))**2 == Order((x - 1)**2, (x, 1))
assert Order(x - 2, (x, 2)) - O(x - 2, (x, 2)) == Order(x - 2, (x, 2))
def test_order_at_infinity():
assert Order(1 + x, (x, oo)) == Order(x, (x, oo))
assert Order(3*x, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo))*3 == Order(x, (x, oo))
assert -28*Order(x, (x, oo)) == Order(x, (x, oo))
assert Order(Order(x, (x, oo)), (x, oo)) == Order(x, (x, oo))
assert Order(Order(x, (x, oo)), (y, oo)) == Order(x, (x, oo), (y, oo))
assert Order(3, (x, oo)) == Order(1, (x, oo))
assert Order(x**2 + x + y, (x, oo)) == O(x**2, (x, oo))
assert Order(x**2 + x + y, (y, oo)) == O(y, (y, oo))
assert Order(2*x, (x, oo))*x == Order(x**2, (x, oo))
assert Order(2*x, (x, oo))/x == Order(1, (x, oo))
assert Order(2*x, (x, oo))*x*exp(1/x) == Order(x**2*exp(1/x), (x, oo))
assert Order(2*x, (x, oo))*x*exp(1/x)/ln(x)**3 == Order(x**2*exp(1/x)*ln(x)**-3, (x, oo))
assert Order(x, (x, oo)) + 1/x == 1/x + Order(x, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo)) + 1 == 1 + Order(x, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo)) + x == x + Order(x, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo)) + x**2 == x**2 + Order(x, (x, oo))
assert Order(1/x, (x, oo)) + 1/x**2 == 1/x**2 + Order(1/x, (x, oo)) == Order(1/x, (x, oo))
assert Order(x, (x, oo)) + exp(1/x) == exp(1/x) + Order(x, (x, oo))
assert Order(x, (x, oo))**2 == Order(x**2, (x, oo))
assert Order(x, (x, oo)) + Order(x**2, (x, oo)) == Order(x**2, (x, oo))
assert Order(x, (x, oo)) + Order(x**-2, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo)) + Order(1/x, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo)) - Order(x, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo)) + Order(1, (x, oo)) == Order(x, (x, oo))
assert Order(x, (x, oo)) + Order(x**2, (x, oo)) == Order(x**2, (x, oo))
assert Order(1/x, (x, oo)) + Order(1, (x, oo)) == Order(1, (x, oo))
assert Order(x, (x, oo)) + Order(exp(1/x), (x, oo)) == Order(x, (x, oo))
assert Order(x**3, (x, oo)) + Order(exp(2/x), (x, oo)) == Order(x**3, (x, oo))
assert Order(x**-3, (x, oo)) + Order(exp(2/x), (x, oo)) == Order(exp(2/x), (x, oo))
# issue 7207
assert Order(exp(x), (x, oo)).expr == Order(2*exp(x), (x, oo)).expr == exp(x)
assert Order(y**x, (x, oo)).expr == Order(2*y**x, (x, oo)).expr == exp(log(y)*x)
def test_add_1():
assert Order(x + x) == Order(x)
assert Order(3*x - 2*x**2) == Order(x)
assert Order(1 + x) == Order(1, x)
assert Order(1 + 1/x) == Order(1/x)
assert Order(ln(x) + 1/ln(x)) == Order(ln(x))
assert Order(exp(1/x) + x) == Order(exp(1/x))
assert Order(exp(1/x) + 1/x**20) == Order(exp(1/x))
def test_contains():
assert Order(1, x) not in Order(1)
assert Order(1) in Order(1, x)
raises(TypeError, lambda: Order(x*y**2) in Order(x**2*y))
def test_contains_4():
assert Order(sin(1/x**2)).contains(Order(cos(1/x**2))) is None
assert Order(cos(1/x**2)).contains(Order(sin(1/x**2))) is None
def test_contains_3():
assert Order(x*y**2).contains(Order(x**2*y)) is None
assert Order(x**2*y).contains(Order(x*y**2)) is None
def test_contains_2():
assert Order(x).contains(Order(y)) is None
assert Order(x).contains(Order(y*x))
assert Order(y*x).contains(Order(x))
assert Order(y).contains(Order(x*y))
assert Order(x).contains(Order(y**2*x))
def test_contains_1():
assert Order(x).contains(Order(x))
assert Order(x).contains(Order(x**2))
assert not Order(x**2).contains(Order(x))
assert not Order(x).contains(Order(1/x))
assert not Order(1/x).contains(Order(exp(1/x)))
assert not Order(x).contains(Order(exp(1/x)))
assert Order(1/x).contains(Order(x))
assert Order(exp(1/x)).contains(Order(x))
assert Order(exp(1/x)).contains(Order(1/x))
assert Order(exp(1/x)).contains(Order(exp(1/x)))
assert Order(exp(2/x)).contains(Order(exp(1/x)))
assert not Order(exp(1/x)).contains(Order(exp(2/x)))
def test_contains_0():
assert Order(1, x).contains(Order(1, x))
assert Order(1, x).contains(Order(1))
assert Order(1).contains(Order(1, x)) is False
def test_order_noncommutative():
A = Symbol('A', commutative=False)
assert Order(A + A*x, x) == Order(1, x)
assert (A + A*x)*Order(x) == Order(x)
assert (A*x)*Order(x) == Order(x**2, x)
assert expand((1 + Order(x))*A*A*x) == A*A*x + Order(x**2, x)
assert expand((A*A + Order(x))*x) == A*A*x + Order(x**2, x)
assert expand((A + Order(x))*A*x) == A*A*x + Order(x**2, x)
def test_free_symbols():
assert Order(1).free_symbols == set()
assert Order(x).free_symbols == {x}
assert Order(1, x).free_symbols == {x}
assert Order(x*y).free_symbols == {x, y}
assert Order(x, x, y).free_symbols == {x, y}
def test_mixing_order_at_zero_and_infinity():
assert (Order(x, (x, 0)) + Order(x, (x, oo))).is_Add
assert Order(x, (x, 0)) + Order(x, (x, oo)) == Order(x, (x, oo)) + Order(x, (x, 0))
assert Order(Order(x, (x, oo))) == Order(x, (x, oo))
# not supported (yet)
raises(NotImplementedError, lambda: Order(x, (x, 0))*Order(x, (x, oo)))
raises(NotImplementedError, lambda: Order(x, (x, oo))*Order(x, (x, 0)))
raises(NotImplementedError, lambda: Order(Order(x, (x, oo)), y))
raises(NotImplementedError, lambda: Order(Order(x), (x, oo)))
def test_multivar_0a():
assert Order(exp(1/x)*exp(1/y)).expr == exp(1/x + 1/y)
def test_issue_9351():
assert exp(x).series(x, 10, 1) == exp(10) + Order(x - 10, (x, 10))
def test_multivar_2():
assert Order(x**2*y + y**2*x, x, y).expr == x**2*y + y**2*x
def test_simple_3():
assert Order(x) + x == Order(x)
assert Order(x) + 2 == 2 + Order(x)
assert Order(x) + x**2 == Order(x)
assert Order(x) + 1/x == 1/x + Order(x)
assert Order(1/x) + 1/x**2 == 1/x**2 + Order(1/x)
assert Order(x) + exp(1/x) == Order(x) + exp(1/x)
def test_multivar_mul_1():
assert Order(x + y)*x == Order(x**2 + y*x, x, y)
def test_simple_5():
assert Order(x) + Order(x**2) == Order(x)
assert Order(x) + Order(x**-2) == Order(x**-2)
assert Order(x) + Order(1/x) == Order(1/x)
def test_multivar_3():
assert (Order(x) + Order(y)).args in [
(Order(x), Order(y)),
(Order(y), Order(x))]
assert Order(x) + Order(y) + Order(x + y) == Order(x + y)
assert (Order(x**2*y) + Order(y**2*x)).args in [
(Order(x*y**2), Order(y*x**2)),
(Order(y*x**2), Order(x*y**2))]
assert (Order(x**2*y) + Order(y*x)) == Order(x*y)
def test_as_expr_variables():
assert Order(x).as_expr_variables(None) == (x, ((x, 0),))
assert Order(x).as_expr_variables((((x, 0),))) == (x, ((x, 0),))
assert Order(y).as_expr_variables(((x, 0),)) == (y, ((x, 0), (y, 0)))
assert Order(y).as_expr_variables(((x, 0), (y, 0))) == (y, ((x, 0), (y, 0)))
def test_simple_1():
o = Rational(0)
assert Order(2*x) == Order(x)
assert Order(x)*3 == Order(x)
assert -28*Order(x) == Order(x)
assert Order(Order(x)) == Order(x)
assert Order(Order(x), y) == Order(Order(x), x, y)
assert Order(-23) == Order(1)
assert Order(exp(x)) == Order(1, x)
assert Order(exp(1/x)).expr == exp(1/x)
assert Order(x*exp(1/x)).expr == x*exp(1/x)
assert Order(x**(o/3)).expr == x**(o/3)
assert Order(x**(5*o/3)).expr == x**(5*o/3)
assert Order(x**2 + x + y, x) == O(1, x)
assert Order(x**2 + x + y, y) == O(1, y)
raises(ValueError, lambda: Order(exp(x), x, x))
raises(TypeError, lambda: Order(x, 2 - x))
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'
beta = Function('beta')
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(FallingFactorial(3,
k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(polylog(x,
y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
assert latex(jacobi(n, a, b,
x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(
n, a, b,
x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a,
x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(
n, a,
x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n,
x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n,
x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a,
x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(
n, a,
x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a,
x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(
n, a,
x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
# Test latex printing of function names with "_"
assert latex(
polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
assert latex(polar_lift(0)**
3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
def test_eval():
assert Order(x).subs(Order(x), 1) == 1
assert Order(x).subs(x, y) == Order(y)
assert Order(x).subs(y, x) == Order(x)
assert Order(x).subs(x, x + y) == Order(x + y, (x, -y))
assert (O(1)**x).is_Pow
def test_order_conjugate_transpose():
x = Symbol('x', real=True)
y = Symbol('y', imaginary=True)
assert conjugate(Order(x)) == Order(conjugate(x))
assert conjugate(Order(y)) == Order(conjugate(y))
assert conjugate(Order(x**2)) == Order(conjugate(x)**2)
assert conjugate(Order(y**2)) == Order(conjugate(y)**2)
assert transpose(Order(x)) == Order(transpose(x))
assert transpose(Order(y)) == Order(transpose(y))
assert transpose(Order(x**2)) == Order(transpose(x)**2)
assert transpose(Order(y**2)) == Order(transpose(y)**2)
def series(self, n=6, coefficient=False, order=True):
"""
Finds the power series expansion of given holonomic function.
Examples
========
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators
>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy import symbols
>>> x = symbols('x')
>>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx')
>>> HolonomicFunction(Dx - 1, x, 0, [1]).series() # e^x
1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)
>>> HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).series(n=8) # sin(x)
x - x**3/6 + x**5/120 - x**7/5040 + O(x**8)
See Also
========
HolonomicFunction.to_sequence
"""
recurrence = self.to_sequence()
l = len(recurrence.u0) - 1
k = recurrence.recurrence.order
x = self.x
seq_dmp = recurrence.recurrence.listofpoly
R = recurrence.recurrence.parent.base
K = R.get_field()
seq = []
if 0 in roots(seq_dmp[-1].rep, filter='Z').keys():
singular = True
else:
singular = False
for i, j in enumerate(seq_dmp):
seq.append(K.new(j.rep))
sub = [-seq[i] / seq[k] for i in range(k)]
sol = [i for i in recurrence.u0]
if l + 1 >= n:
pass
else:
# use the initial conditions to find the next term
for i in range(l + 1 - k, n - k):
coeff = S(0)
for j in range(k):
if i + j >= 0:
coeff += DMFsubs(sub[j], i) * sol[i + j]
sol.append(coeff)
if coefficient:
return sol
ser = S(0)
for i, j in enumerate(sol):
ser += x**i * j
if order:
return ser + Order(x**n, x)
else:
return ser
