def test_jacobi():
n = Symbol("n")
a = Symbol("a")
b = Symbol("b")
assert jacobi(0, a, b, x) == 1
assert jacobi(1, a, b, x) == a/2 - b/2 + x*(a/2 + b/2 + 1)
assert jacobi(n, a, a, x) == RisingFactorial(a + 1, n)*gegenbauer(n, a + S(1)/2, x)/RisingFactorial(2*a + 1, n)
assert jacobi(n, a, -a, x) == ((-1)**a*(-x + 1)**(-a/2)*(x + 1)**(a/2)*assoc_legendre(n, a, x)*
factorial(-a + n)*gamma(a + n + 1)/(factorial(a + n)*gamma(n + 1)))
assert jacobi(n, -b, b, x) == ((-x + 1)**(b/2)*(x + 1)**(-b/2)*assoc_legendre(n, b, x)*
gamma(-b + n + 1)/gamma(n + 1))
assert jacobi(n, 0, 0, x) == legendre(n, x)
assert jacobi(n, S.Half, S.Half, x) == RisingFactorial(S(3)/2, n)*chebyshevu(n, x)/factorial(n + 1)
assert jacobi(n, -S.Half, -S.Half, x) == RisingFactorial(S(1)/2, n)*chebyshevt(n, x)/factorial(n)
X = jacobi(n, a, b, x)
assert isinstance(X, jacobi)
assert jacobi(n, a, b, -x) == (-1)**n*jacobi(n, b, a, x)
assert jacobi(n, a, b, 0) == 2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1))
assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n)/factorial(n)
m = Symbol("m", positive=True)
assert jacobi(m, a, b, oo) == oo*RisingFactorial(a + b + m + 1, m)
assert conjugate(jacobi(m, a, b, x)) == jacobi(m, conjugate(a), conjugate(b), conjugate(x))
assert diff(jacobi(n,a,b,x), n) == Derivative(jacobi(n, a, b, x), n)
assert diff(jacobi(n,a,b,x), x) == (a/2 + b/2 + n/2 + S(1)/2)*jacobi(n - 1, a + 1, b + 1, x)
def test_manualintegrate_orthogonal_poly():
n = symbols('n')
a, b = 7, S(5)/3
polys = [jacobi(n, a, b, x), gegenbauer(n, a, x), chebyshevt(n, x),
chebyshevu(n, x), legendre(n, x), hermite(n, x), laguerre(n, x),
assoc_laguerre(n, a, x)]
for p in polys:
integral = manualintegrate(p, x)
for deg in [-2, -1, 0, 1, 3, 5, 8]:
# some accept negative "degree", some do not
try:
p_subbed = p.subs(n, deg)
except ValueError:
continue
assert (integral.subs(n, deg).diff(x) - p_subbed).expand() == 0
# can also integrate simple expressions with these polynomials
q = x*p.subs(x, 2*x + 1)
integral = manualintegrate(q, x)
for deg in [2, 4, 7]:
assert (integral.subs(n, deg).diff(x) - q.subs(n, deg)).expand() == 0
# cannot integrate with respect to any other parameter
t = symbols('t')
for i in range(len(p.args) - 1):
new_args = list(p.args)
new_args[i] = t
assert isinstance(manualintegrate(p.func(*new_args), t), Integral)
def test_manualintegrate_orthogonal_poly():
n = symbols("n")
a, b = 7, Rational(5, 3)
polys = [
jacobi(n, a, b, x),
gegenbauer(n, a, x),
chebyshevt(n, x),
chebyshevu(n, x),
legendre(n, x),
hermite(n, x),
laguerre(n, x),
assoc_laguerre(n, a, x),
]
for p in polys:
integral = manualintegrate(p, x)
for deg in [-2, -1, 0, 1, 3, 5, 8]:
# some accept negative "degree", some do not
try:
p_subbed = p.subs(n, deg)
except ValueError:
continue
assert (integral.subs(n, deg).diff(x) - p_subbed).expand() == 0
# can also integrate simple expressions with these polynomials
q = x * p.subs(x, 2 * x + 1)
integral = manualintegrate(q, x)
for deg in [2, 4, 7]:
assert (integral.subs(n, deg).diff(x) - q.subs(n, deg)).expand() == 0
# cannot integrate with respect to any other parameter
t = symbols("t")
for i in range(len(p.args) - 1):
new_args = list(p.args)
new_args[i] = t
assert isinstance(manualintegrate(p.func(*new_args), t), Integral)
def callback(q, v=0, u = 0, w=0, kind = 0):
ans = ''
if kind == 1:
ans = str(sp.gamma(v))
elif kind == 2:
ans = str(sp.gamma(u) * sp.gamma(v) / sp.gamma(u + v))
elif kind == 3:
ans = str(functions.Legendre_Polynomials(v))
elif kind == 4:
ans = str(sp.assoc_legendre(v, u, x))
elif kind == 5:
ans = str(functions.bessel_function_1st(v))
elif kind == 6:
ans = str(sp.jacobi(u, v, w, x))
elif kind == 7:
ans = str(sp.jacobi_normalized(u, v, w, x))
elif kind == 8:
ans = str(sp.gegenbauer(u, v, x))
elif kind == 9:
# 1st kind
ans = str(sp.chebyshevt(u, x))
elif kind == 10:
ans = str(sp.chebyshevt_root(u, v))
elif kind == 11:
# 2nd kind
ans = str(sp.chebyshevu(u, x))
elif kind == 12:
ans = str(sp.chebyshevu_root(u, v))
elif kind == 13:
ans = str(sp.hermite(u, x))
elif kind == 14:
ans = str(sp.laguerre(u, x))
elif kind == 15:
ans = str(sp.assoc_laguerre(u, v, x))
q.put(ans)
def test_jacobi():
n = Symbol("n")
a = Symbol("a")
b = Symbol("b")
assert jacobi(0, a, b, x) == 1
assert jacobi(1, a, b, x) == a/2 - b/2 + x*(a/2 + b/2 + 1)
assert jacobi(n, a, a, x) == RisingFactorial(
a + 1, n)*gegenbauer(n, a + S.Half, x)/RisingFactorial(2*a + 1, n)
assert jacobi(n, a, -a, x) == ((-1)**a*(-x + 1)**(-a/2)*(x + 1)**(a/2)*assoc_legendre(n, a, x)*
factorial(-a + n)*gamma(a + n + 1)/(factorial(a + n)*gamma(n + 1)))
assert jacobi(n, -b, b, x) == ((-x + 1)**(b/2)*(x + 1)**(-b/2)*assoc_legendre(n, b, x)*
gamma(-b + n + 1)/gamma(n + 1))
assert jacobi(n, 0, 0, x) == legendre(n, x)
assert jacobi(n, S.Half, S.Half, x) == RisingFactorial(
Rational(3, 2), n)*chebyshevu(n, x)/factorial(n + 1)
assert jacobi(n, Rational(-1, 2), Rational(-1, 2), x) == RisingFactorial(
S.Half, n)*chebyshevt(n, x)/factorial(n)
X = jacobi(n, a, b, x)
assert isinstance(X, jacobi)
assert jacobi(n, a, b, -x) == (-1)**n*jacobi(n, b, a, x)
assert jacobi(n, a, b, 0) == 2**(-n)*gamma(a + n + 1)*hyper(
(-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1))
assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n)/factorial(n)
m = Symbol("m", positive=True)
assert jacobi(m, a, b, oo) == oo*RisingFactorial(a + b + m + 1, m)
assert unchanged(jacobi, n, a, b, oo)
assert conjugate(jacobi(m, a, b, x)) == \
jacobi(m, conjugate(a), conjugate(b), conjugate(x))
_k = Dummy('k')
assert diff(jacobi(n, a, b, x), n) == Derivative(jacobi(n, a, b, x), n)
assert diff(jacobi(n, a, b, x), a).dummy_eq(Sum((jacobi(n, a, b, x) +
(2*_k + a + b + 1)*RisingFactorial(_k + b + 1, -_k + n)*jacobi(_k, a,
b, x)/((-_k + n)*RisingFactorial(_k + a + b + 1, -_k + n)))/(_k + a
+ b + n + 1), (_k, 0, n - 1)))
assert diff(jacobi(n, a, b, x), b).dummy_eq(Sum(((-1)**(-_k + n)*(2*_k +
a + b + 1)*RisingFactorial(_k + a + 1, -_k + n)*jacobi(_k, a, b, x)/
((-_k + n)*RisingFactorial(_k + a + b + 1, -_k + n)) + jacobi(n, a,
b, x))/(_k + a + b + n + 1), (_k, 0, n - 1)))
assert diff(jacobi(n, a, b, x), x) == \
(a/2 + b/2 + n/2 + S.Half)*jacobi(n - 1, a + 1, b + 1, x)
assert jacobi_normalized(n, a, b, x) == \
(jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)
/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))))
raises(ValueError, lambda: jacobi(-2.1, a, b, x))
raises(ValueError, lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
assert jacobi(n, a, b, x).rewrite("polynomial").dummy_eq(Sum((S.Half - x/2)
**_k*RisingFactorial(-n, _k)*RisingFactorial(_k + a + 1, -_k + n)*
RisingFactorial(a + b + n + 1, _k)/factorial(_k), (_k, 0, n))/factorial(n))
raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
def test_jacobi():
n = Symbol("n")
a = Symbol("a")
b = Symbol("b")
assert jacobi(0, a, b, x) == 1
assert jacobi(1, a, b, x) == a / 2 - b / 2 + x * (a / 2 + b / 2 + 1)
assert jacobi(n, a, a, x) == RisingFactorial(a + 1, n) * gegenbauer(
n, a + S(1) / 2, x) / RisingFactorial(2 * a + 1, n)
assert jacobi(n, a, -a,
x) == ((-1)**a * (-x + 1)**(-a / 2) * (x + 1)**(a / 2) *
assoc_legendre(n, a, x) * factorial(-a + n) *
gamma(a + n + 1) / (factorial(a + n) * gamma(n + 1)))
assert jacobi(n, -b, b, x) == ((-x + 1)**(b / 2) * (x + 1)**(-b / 2) *
assoc_legendre(n, b, x) *
gamma(-b + n + 1) / gamma(n + 1))
assert jacobi(n, 0, 0, x) == legendre(n, x)
assert jacobi(n, S.Half,
S.Half, x) == RisingFactorial(S(3) / 2, n) * chebyshevu(
n, x) / factorial(n + 1)
assert jacobi(
n, -S.Half, -S.Half,
x) == RisingFactorial(S(1) / 2, n) * chebyshevt(n, x) / factorial(n)
X = jacobi(n, a, b, x)
assert isinstance(X, jacobi)
assert jacobi(n, a, b, -x) == (-1)**n * jacobi(n, b, a, x)
assert jacobi(n, a, b, 0) == 2**(-n) * gamma(a + n + 1) * hyper(
(-b - n, -n), (a + 1, ), -1) / (factorial(n) * gamma(a + 1))
assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n) / factorial(n)
m = Symbol("m", positive=True)
assert jacobi(m, a, b, oo) == oo * RisingFactorial(a + b + m + 1, m)
assert conjugate(jacobi(m, a, b, x)) == \
jacobi(m, conjugate(a), conjugate(b), conjugate(x))
assert diff(jacobi(n, a, b, x), n) == Derivative(jacobi(n, a, b, x), n)
assert diff(jacobi(n, a, b, x), x) == \
(a/2 + b/2 + n/2 + S(1)/2)*jacobi(n - 1, a + 1, b + 1, x)
assert jacobi_normalized(n, a, b, x) == \
(jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)
/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))))
raises(ValueError, lambda: jacobi(-2.1, a, b, x))
raises(ValueError,
lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
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"\lvert{x}\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 pol(self, t):
c = ((self.n + self.k) * (self.n + self.k - 1)) / (2 *
(2 * self.k - 1))
d = (self.k + 0.5)
return c * gegenbauer(self.n - self.k, self.k - 0.5, t) + d * (
1 - t * t) * gegenbauer(self.n - self.k - 2, self.k + 1.5, t)
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"\lvert{x}\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_gegenbauer():
n = Symbol("n")
a = Symbol("a")
assert gegenbauer(0, a, x) == 1
assert gegenbauer(1, a, x) == 2*a*x
assert gegenbauer(2, a, x) == -a + x**2*(2*a**2 + 2*a)
assert gegenbauer(3, a, x) == \
x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
assert gegenbauer(-1, a, x) == 0
assert gegenbauer(n, S(1)/2, x) == legendre(n, x)
assert gegenbauer(n, 1, x) == chebyshevu(n, x)
assert gegenbauer(n, -1, x) == 0
X = gegenbauer(n, a, x)
assert isinstance(X, gegenbauer)
assert gegenbauer(n, a, -x) == (-1)**n*gegenbauer(n, a, x)
assert gegenbauer(n, a, 0) == 2**n*sqrt(pi) * \
gamma(a + n/2)/(gamma(a)*gamma(-n/2 + S(1)/2)*gamma(n + 1))
assert gegenbauer(n, a, 1) == gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
assert gegenbauer(n, Rational(3, 4), -1) == zoo
m = Symbol("m", positive=True)
assert gegenbauer(m, a, oo) == oo*RisingFactorial(a, m)
assert conjugate(gegenbauer(n, a, x)) == gegenbauer(n, conjugate(a), conjugate(x))
assert diff(gegenbauer(n, a, x), n) == Derivative(gegenbauer(n, a, x), n)
assert diff(gegenbauer(n, a, x), x) == 2*a*gegenbauer(n - 1, a + 1, 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)) == 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)"
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(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 )}', latex(Chi(x)**2)
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_gegenbauer():
n = Symbol("n")
a = Symbol("a")
assert gegenbauer(0, a, x) == 1
assert gegenbauer(1, a, x) == 2 * a * x
assert gegenbauer(2, a, x) == -a + x**2 * (2 * a**2 + 2 * a)
assert gegenbauer(3, a, x) == \
x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
assert gegenbauer(-1, a, x) == 0
assert gegenbauer(n, S(1) / 2, x) == legendre(n, x)
assert gegenbauer(n, 1, x) == chebyshevu(n, x)
assert gegenbauer(n, -1, x) == 0
X = gegenbauer(n, a, x)
assert isinstance(X, gegenbauer)
assert gegenbauer(n, a, -x) == (-1)**n * gegenbauer(n, a, x)
assert gegenbauer(n, a, 0) == 2**n*sqrt(pi) * \
gamma(a + n/2)/(gamma(a)*gamma(-n/2 + S(1)/2)*gamma(n + 1))
assert gegenbauer(n, a,
1) == gamma(2 * a + n) / (gamma(2 * a) * gamma(n + 1))
assert gegenbauer(n, Rational(3, 4), -1) == zoo
m = Symbol("m", positive=True)
assert gegenbauer(m, a, oo) == oo * RisingFactorial(a, m)
assert conjugate(gegenbauer(n,
a, x)) == gegenbauer(n, conjugate(a),
conjugate(x))
assert diff(gegenbauer(n, a, x), n) == Derivative(gegenbauer(n, a, x), n)
assert diff(gegenbauer(n, a, x), x) == 2 * a * gegenbauer(n - 1, a + 1, x)
def test_gegenbauer():
n = Symbol("n")
a = Symbol("a")
assert gegenbauer(0, a, x) == 1
assert gegenbauer(1, a, x) == 2*a*x
assert gegenbauer(2, a, x) == -a + x**2*(2*a**2 + 2*a)
assert gegenbauer(3, a, x) == \
x**3*(4*a**3/3 + 4*a**2 + a*Rational(8, 3)) + x*(-2*a**2 - 2*a)
assert gegenbauer(-1, a, x) == 0
assert gegenbauer(n, S.Half, x) == legendre(n, x)
assert gegenbauer(n, 1, x) == chebyshevu(n, x)
assert gegenbauer(n, -1, x) == 0
X = gegenbauer(n, a, x)
assert isinstance(X, gegenbauer)
assert gegenbauer(n, a, -x) == (-1)**n*gegenbauer(n, a, x)
assert gegenbauer(n, a, 0) == 2**n*sqrt(pi) * \
gamma(a + n/2)/(gamma(a)*gamma(-n/2 + S.Half)*gamma(n + 1))
assert gegenbauer(n, a, 1) == gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
assert gegenbauer(n, Rational(3, 4), -1) is zoo
assert gegenbauer(n, Rational(1, 4), -1) == (sqrt(2)*cos(pi*(n + S.One/4))*
gamma(n + S.Half)/(sqrt(pi)*gamma(n + 1)))
m = Symbol("m", positive=True)
assert gegenbauer(m, a, oo) == oo*RisingFactorial(a, m)
assert unchanged(gegenbauer, n, a, oo)
assert conjugate(gegenbauer(n, a, x)) == gegenbauer(n, conjugate(a), conjugate(x))
_k = Dummy('k')
assert diff(gegenbauer(n, a, x), n) == Derivative(gegenbauer(n, a, x), n)
assert diff(gegenbauer(n, a, x), a).dummy_eq(Sum((2*(-1)**(-_k + n) + 2)*
(_k + a)*gegenbauer(_k, a, x)/((-_k + n)*(_k + 2*a + n)) + ((2*_k +
2)/((_k + 2*a)*(2*_k + 2*a + 1)) + 2/(_k + 2*a + n))*gegenbauer(n, a
, x), (_k, 0, n - 1)))
assert diff(gegenbauer(n, a, x), x) == 2*a*gegenbauer(n - 1, a + 1, x)
assert gegenbauer(n, a, x).rewrite('polynomial').dummy_eq(
Sum((-1)**_k*(2*x)**(-2*_k + n)*RisingFactorial(a, -_k + n)
/(factorial(_k)*factorial(-2*_k + n)), (_k, 0, floor(n/2))))
raises(ArgumentIndexError, lambda: gegenbauer(n, a, x).fdiff(4))
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}'