def test_euler_polynomials():
assert euler(0, x) == 1
assert euler(1, x) == x - Rational(1, 2)
assert euler(2, x) == x**2 - x
assert euler(3, x) == x**3 - (3*x**2)/2 + Rational(1, 4)
m = Symbol('m')
assert isinstance(euler(m, x), euler)
from sympy import Float
A = Float('-0.46237208575048694923364757452876131e8') # from Maple
B = euler(19, S.Pi.evalf(32))
assert abs((A - B)/A) < 1e-31 # expect low relative error
C = euler(19, S.Pi, evaluate=False).evalf(32)
assert abs((A - C)/A) < 1e-31
def test_euler_polynomials():
assert euler(0, x) == 1
assert euler(1, x) == x - Rational(1, 2)
assert euler(2, x) == x**2 - x
assert euler(3, x) == x**3 - (3 * x**2) / 2 + Rational(1, 4)
m = Symbol('m')
assert isinstance(euler(m, x), euler)
from sympy import Float
A = Float('-0.46237208575048694923364757452876131e8') # from Maple
B = euler(19, S.Pi.evalf(32))
assert abs((A - B) / A) < 1e-31 # expect low relative error
C = euler(19, S.Pi, evaluate=False).evalf(32)
assert abs((A - C) / A) < 1e-31
def test_euler_polynomial_rewrite():
m = Symbol('m')
A = euler(m, x).rewrite('Sum')
assert A.subs({m: 3, x: 5}).doit() == euler(3, 5)
def test_euler_odd():
n = Symbol('n', odd=True, positive=True)
assert euler(n) == 0
n = Symbol('n', odd=True)
assert euler(n) != 0
def test_euler_failing():
# depends on dummy variables being implemented https://github.com/sympy/sympy/issues/5665
assert euler(2 * n).rewrite(Sum) == I * Sum(
Sum((-1)**_j * 2**(-_k) * I**(-_k) *
(-2 * _j + _k)**(2 * n + 1) * binomial(_k, _j) / _k, (_j, 0, _k)),
(_k, 1, 2 * n + 1))
def test_euler():
assert euler(0) == 1
assert euler(1) == 0
assert euler(2) == -1
assert euler(3) == 0
assert euler(4) == 5
assert euler(6) == -61
assert euler(8) == 1385
assert euler(20, evaluate=False) != 370371188237525
n = Symbol('n', integer=True)
assert euler(n) != -1
assert euler(n).subs(n, 2) == -1
raises(ValueError, lambda: euler(-2))
raises(ValueError, lambda: euler(-3))
raises(ValueError, lambda: euler(2.3))
assert euler(20).evalf() == 370371188237525.0
assert euler(20, evaluate=False).evalf() == 370371188237525.0
assert euler(n).rewrite(Sum) == euler(n)
# XXX: Not sure what the guy who wrote this test was trying to do with the _j and _k stuff
n = Symbol('n', integer=True, nonnegative=True)
assert euler(2 * n + 1).rewrite(Sum) == 0
def test_euler():
assert euler(0) == 1
assert euler(1) == 0
assert euler(2) == -1
assert euler(3) == 0
assert euler(4) == 5
assert euler(6) == -61
assert euler(8) == 1385
assert euler(20, evaluate=False) != 370371188237525
n = Symbol('n', integer=True)
assert euler(n) != -1
assert euler(n).subs(n, 2) == -1
raises(ValueError, lambda: euler(-2))
raises(ValueError, lambda: euler(-3))
raises(ValueError, lambda: euler(2.3))
assert euler(20).evalf() == 370371188237525.0
assert euler(20, evaluate=False).evalf() == 370371188237525.0
assert euler(n).rewrite(Sum) == euler(n)
n = Symbol('n', integer=True, nonnegative=True)
assert euler(2 * n + 1).rewrite(Sum) == 0
_j = Dummy('j')
_k = Dummy('k')
assert euler(2 * n).rewrite(Sum).dummy_eq(I * Sum(
(-1)**_j * 2**(-_k) * I**(-_k) *
(-2 * _j + _k)**(2 * n + 1) * binomial(_k, _j) / _k, (_j, 0, _k),
(_k, 1, 2 * n + 1)))
def test_euler_sequence():
a = euler_sequence(20)
for n, b in enumerate(a):
assert b == abs(euler(2 * n))
def test_euler_failing():
# depends on dummy variables being implemented https://github.com/sympy/sympy/issues/5665
assert euler(2*n).rewrite(Sum) == I*Sum(Sum((-1)**_j*2**(-_k)*I**(-_k)*(-2*_j + _k)**(2*n + 1)*binomial(_k, _j)/_k, (_j, 0, _k)), (_k, 1, 2*n + 1))
def test_euler():
assert euler(0) == 1
assert euler(1) == 0
assert euler(2) == -1
assert euler(3) == 0
assert euler(4) == 5
assert euler(6) == -61
assert euler(8) == 1385
assert euler(20, evaluate=False) != 370371188237525
n = Symbol('n', integer=True)
assert euler(n) != -1
assert euler(n).subs(n, 2) == -1
assert euler(20).evalf() == 370371188237525.0
assert euler(20, evaluate=False).evalf() == 370371188237525.0
assert euler(n).rewrite(Sum) == euler(n)
# XXX: Not sure what the guy who wrote this test was trying to do with the _j and _k stuff
assert euler(2*n + 1).rewrite(Sum) == 0
def test_issue_8496():
n = Symbol("n")
k = Symbol("k")
raises(TypeError, lambda: catalan(n, k))
raises(TypeError, lambda: euler(n, k))
def test_issue_8496():
n = Symbol("n")
k = Symbol("k")
raises(TypeError, lambda: catalan(n, k))
raises(TypeError, lambda: euler(n, k))
def test_euler_polynomial_rewrite():
m = Symbol('m')
A = euler(m, x).rewrite('Sum');
assert A.subs({m:3, x:5}).doit() == euler(3, 5)
def test_euler_odd():
n = Symbol('n', odd=True, positive=True)
assert euler(n) == 0
n = Symbol('n', odd=True)
assert euler(n) != 0
