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_sympy__functions__special__polynomials__hermite():
from sympy.functions.special.polynomials import hermite
assert _test_args(hermite(x, 2))
def test_hermite():
assert hermite(0, x) == 1
assert hermite(1, x) == 2 * x
assert hermite(2, x) == 4 * x**2 - 2
assert hermite(3, x) == 8 * x**3 - 12 * x
assert hermite(4, x) == 16 * x**4 - 48 * x**2 + 12
assert hermite(6, x) == 64 * x**6 - 480 * x**4 + 720 * x**2 - 120
n = Symbol("n")
assert unchanged(hermite, n, x)
assert hermite(n, -x) == (-1)**n * hermite(n, x)
assert unchanged(hermite, -n, x)
assert hermite(n, 0) == 2**n * sqrt(pi) / gamma(S.Half - n / 2)
assert hermite(n, oo) is oo
assert conjugate(hermite(n, x)) == hermite(n, conjugate(x))
_k = Dummy('k')
assert hermite(n, x).rewrite("polynomial").dummy_eq(
factorial(n) * Sum((-1)**_k * (2 * x)**(-2 * _k + n) /
(factorial(_k) * factorial(-2 * _k + n)),
(_k, 0, floor(n / 2))))
assert diff(hermite(n, x), x) == 2 * n * hermite(n - 1, x)
assert diff(hermite(n, x), n) == Derivative(hermite(n, x), n)
raises(ArgumentIndexError, lambda: hermite(n, x).fdiff(3))
def test_sympy__functions__special__polynomials__hermite():
from sympy.functions.special.polynomials import hermite
assert _test_args(hermite(x, 2))