def integralKernelMinus(self, n):
n = n - 1
lala1 = sy.jacobi(n, si.Rational(1, 2), si.Rational(3, 2), r[0])
lala2 = sy.jacobi(n, si.Rational(3, 2), si.Rational(1, 2), r[0])
return self.prefactor(n) * (si.cos(r[0] / 2) * lala1 * ident +
I * si.sin(r[0] / 2) * lala2 * sigma3)
def gen(i_deg):
l_xi1 = sympy.symbols('xi_1') # first tetrahedral coord
l_xi2 = sympy.symbols('xi_2') # second tetrahedral coord
l_xi3 = sympy.symbols('xi_3') # third tetrahedral coord
l_eta1 = sympy.symbols('eta_1') # first collapsed coord
l_eta2 = sympy.symbols('eta_2') # second collapsed coord
l_eta3 = sympy.symbols('eta_3') # third collapsed coord
# basis
l_basis = []
# derive basis functions
for l_de in range(0, i_deg + 1):
for l_p1 in range(0, l_de + 1):
# first principal function
l_psiA = sympy.jacobi(l_p1, 0, 0, l_eta1)
for l_p2 in range(0, l_de + 1):
# second principal function
l_psiB = (((1 - l_eta2) / 2)**l_p1 *
sympy.jacobi(l_p2, 2 * l_p1 + 1, 0, l_eta2))
for l_p3 in range(0, l_de + 1):
if (
l_p1 + l_p2 + l_p3 == l_de
): # build hierarchical basis following pascals triangle
# third principal function
l_psiC = (
(1 - l_eta3) / 2)**(l_p1 + l_p2) * sympy.jacobi(
l_p3, 2 * l_p1 + 2 * l_p2 + 2, 0, l_eta3)
l_basis = l_basis + [l_psiA * l_psiB * l_psiC]
# insert tetrahedral coordinates
l_basis[-1] = l_basis[-1].subs(
l_eta1, 2 * (1 + l_xi1) / (-l_xi2 - l_xi3) - 1)
l_basis[-1] = l_basis[-1].subs(
l_eta2, 2 * (1 + l_xi2) / (1 - l_xi3) - 1)
l_basis[-1] = l_basis[-1].subs(l_eta3, l_xi3)
# use tets with xi1, xi2, xi3 in [0,1]
l_basis[-1] = l_basis[-1].subs(l_xi1, 2 * l_xi1 - 1)
l_basis[-1] = l_basis[-1].subs(l_xi2, 2 * l_xi2 - 1)
l_basis[-1] = l_basis[-1].subs(l_xi3, 2 * l_xi3 - 1)
# simplify basis
l_basis[-1] = sympy.simplify(l_basis[-1])
# cancel denominators
l_basis[-1] = sympy.cancel(l_basis[-1])
return [l_xi1, l_xi2, l_xi3], l_basis
def scalar(i, j):
x = sympy.symbols('x')
wj = sympy.jacobi(j, 1, 1, x) * (1 - np.power(x, 2))
d = sympy.lambdify(x, sympy.diff(wj, x, 1))
d2 = sympy.lambdify(x, sympy.diff(wj, x, 2))
l = integrate.quad(
lambda x:
(-1 * d2(x) / (x - 3) +
(1 + x / 2) * d(x) + sympy.exp(x / 2) * sympy.jacobi(j, 1, 1, x) *
(1 - np.power(x, 2))) * sympy.jacobi(i, 1, 1, x) *
(1 - np.power(x, 2)), -1, 1)[0]
return l
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 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 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 jacobi(self, x, alpha, beta, N):
V = np.zeros((x.shape[0], N))
if mode == 'numpy':
for n in range(N):
V[:, n] = eval_jacobi(n, alpha, beta, x)
else:
for n in range(N):
V[:, n] = sp.lambdify(xp, sp.jacobi(n, alpha, beta, xp),
'mpmath')(x)
return V
def jacobi(self, x, alpha, beta, N):
V = np.zeros((x.shape[0], N))
if mode == 'numpy':
for n in range(N):
V[:, n] = eval_jacobi(n, alpha, beta, x)
else:
X = sp.symbols('x', real=True)
for n in range(N):
V[:, n] = sp.lambdify(X, sp.jacobi(n, alpha, beta, X),
'mpmath')(x)
return V
def galerkin(n, w):
A = np.eye(n)
B = np.ones((n, 1))
for i in range(n):
B[i] = integrate.quad(
lambda x: (2 - x) * sympy.jacobi(i, 1, 1, x) *
(1 - np.power(x, 2)), -1, 1)[0]
for j in range(n):
A[i][j] = scalar(i, j)
C = np.linalg.solve(A, B)
mu = infnorm(A) * infnorm(np.linalg.inv(A))
return A, B, C, mu
def evaluate_basis_derivative(self, x=None, i=0, k=0, output_array=None):
if x is None:
x = self.points_and_weights(mode=mode)[0]
#x = np.atleast_1d(x)
if output_array is None:
output_array = np.zeros(x.shape, dtype=self.dtype)
if mode == 'numpy':
dj = np.prod(
np.array(
[i + self.alpha + self.beta + 1 + j for j in range(k)]))
output_array[:] = dj / 2**k * eval_jacobi(i - k, self.alpha + k,
self.beta + k, x)
else:
f = sp.jacobi(i, self.alpha, self.beta, xp)
output_array[:] = sp.lambdify(xp, f.diff(xp, k), 'mpmath')(x)
return output_array
def test_JacobiPolys(self):
n = 10
m = 10
listn = np.zeros(n, dtype=object)
vals = np.zeros((n, m), dtype=object)
for i in range(n):
#print( sy.simplify(JacPol(sy.Rational(1,2),sy.Rational(3,2), i)),'=?=', sy.jacobi(i, 1/2,3/2, r[0]))
listn[i] = si.lambdify(r[0], [
sy.simplify(JacPol(sy.Rational(1, 2), sy.Rational(3, 2), i)) -
sy.jacobi(i, sy.Rational(1, 2), sy.Rational(3, 2), r[0])
])
for i in range(n):
for j in range(m):
vals[i, j] = listn[i](j)[0] #, maxn = 10)
print(vals)
np.testing.assert_almost_equal(vals, np.zeros((n, m)), 10)
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))
"Проекционные методы решения краевой задачи для обыкновенного дифференциального уравнения второго порядка"
)
print("Вариант 3")
v = int(
input(
"Введите число координатных функций или нажмите 0, чтобы оставить значения от 3 до 10:"
))
CondGalerkin = []
CondColloc = []
if v == 0:
for n in range(3, 11):
w = [
] # формируем семейство ортогональных функций — здесь это многочлены Якоби
for i in range(n):
w.append(sympy.jacobi(i, 1, 1, x) * (1 - np.power(x, 2)))
A, B, C, mu = galerkin(n, w)
A1, B1, C1, mu1 = collocation(n, w)
CondGalerkin.append([
n, mu,
solution(-0.5, C),
solution(0, C),
solution(0.5, C),
np.abs(solution(-0.5, C) - solution(-0.5, C1))
])
CondColloc.append([
n, mu1,
solution(-0.5, C1),
solution(0, C1),
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 sympy_basis(self, i=0, x=sp.symbols('x', real=True)):
return (1 - x**2)**3 * sp.jacobi(i, 3, 3, x)
def sympy_basis(self, i=0, x=sp.symbols('x', real=True)):
return sp.jacobi(i, self.alpha, self.beta, 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)}"
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 sympy_basis(self, i=0):
x = sp.symbols('x')
return (1 - x**2)**3 * sp.jacobi(i, 3, 3, x)
def sympy_basis(self, i=0, x=xp):
return sp.jacobi(i, self.alpha, self.beta, x)
def sympy_basis(self, i=0):
x = sp.symbols('x')
return sp.jacobi(i, self.alpha, self.beta, x)
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_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 radialFunc():
ma = sp.Abs(m)
p = (n - ma) / 2
_rhs = ((-1)**p) * (rho**ma) * sp.jacobi(p, ma, 0, 1 - 2 * rho**2)
_lhs = Rnm
return sp.Eq(_lhs, _rhs)
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 sympy_basis(self, i=0, x=xp):
return (1 - x**2)**3 * sp.jacobi(i, 3, 3, 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"\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 ep(i, d):
f = diff(ks(i, d) * ((symcos(t)) ** d) * jacobi(i, 0.5 * d - 1, 0.5 * d, symcos(2 * t)), t)
return lambdify(t, f)