def test_chebyshev():
assert chebyshevt(0, x) == 1
assert chebyshevt(1, x) == x
assert chebyshevt(2, x) == 2*x**2 - 1
assert chebyshevt(3, x) == 4*x**3 - 3*x
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert chebyshevt(n, z) == 0
raises(ValueError, lambda: chebyshevt_root(n, n))
for n in range(1, 4):
for k in range(n):
z = chebyshevu_root(n, k)
assert chebyshevu(n, z) == 0
raises(ValueError, lambda: chebyshevu_root(n, n))
n = Symbol("n")
X = chebyshevt(n, x)
assert isinstance(X, chebyshevt)
assert unchanged(chebyshevt, n, x)
assert chebyshevt(n, -x) == (-1)**n*chebyshevt(n, x)
assert chebyshevt(-n, x) == chebyshevt(n, x)
assert chebyshevt(n, 0) == cos(pi*n/2)
assert chebyshevt(n, 1) == 1
assert chebyshevt(n, oo) is oo
assert conjugate(chebyshevt(n, x)) == chebyshevt(n, conjugate(x))
assert diff(chebyshevt(n, x), x) == n*chebyshevu(n - 1, x)
X = chebyshevu(n, x)
assert isinstance(X, chebyshevu)
y = Symbol('y')
assert chebyshevu(n, -x) == (-1)**n*chebyshevu(n, x)
assert chebyshevu(-n, x) == -chebyshevu(n - 2, x)
assert unchanged(chebyshevu, -n + y, x)
assert chebyshevu(n, 0) == cos(pi*n/2)
assert chebyshevu(n, 1) == n + 1
assert chebyshevu(n, oo) is oo
assert conjugate(chebyshevu(n, x)) == chebyshevu(n, conjugate(x))
assert diff(chebyshevu(n, x), x) == \
(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
_k = Dummy('k')
assert chebyshevt(n, x).rewrite("polynomial").dummy_eq(Sum(x**(-2*_k + n)
*(x**2 - 1)**_k*binomial(n, 2*_k), (_k, 0, floor(n/2))))
assert chebyshevu(n, x).rewrite("polynomial").dummy_eq(Sum((-1)**_k*(2*x)
**(-2*_k + n)*factorial(-_k + n)/(factorial(_k)*
factorial(-2*_k + n)), (_k, 0, floor(n/2))))
raises(ArgumentIndexError, lambda: chebyshevt(n, x).fdiff(1))
raises(ArgumentIndexError, lambda: chebyshevt(n, x).fdiff(3))
raises(ArgumentIndexError, lambda: chebyshevu(n, x).fdiff(1))
raises(ArgumentIndexError, lambda: chebyshevu(n, x).fdiff(3))
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_chebyshev():
raises(ValueError, lambda: chebyshevt(-1, x))
raises(ValueError, lambda: chebyshevu(-1, x))
assert chebyshevt(0, x) == 1
assert chebyshevt(1, x) == x
assert chebyshevt(2, x) == 2*x**2-1
assert chebyshevt(3, x) == 4*x**3-3*x
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert chebyshevt(n, z) == 0
raises(ValueError, lambda: chebyshevt_root(n, n))
for n in range(1, 4):
for k in range(n):
z = chebyshevu_root(n, k)
assert chebyshevu(n, z) == 0
raises(ValueError, lambda: chebyshevu_root(n, n))
def test_chebyshev():
raises(ValueError, lambda: chebyshevt(-1, x))
raises(ValueError, lambda: chebyshevu(-1, x))
assert chebyshevt(0, x) == 1
assert chebyshevt(1, x) == x
assert chebyshevt(2, x) == 2 * x**2 - 1
assert chebyshevt(3, x) == 4 * x**3 - 3 * x
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert chebyshevt(n, z) == 0
raises(ValueError, lambda: chebyshevt_root(n, n))
for n in range(1, 4):
for k in range(n):
z = chebyshevu_root(n, k)
assert chebyshevu(n, z) == 0
raises(ValueError, lambda: chebyshevu_root(n, n))
def test_chebyshev():
assert chebyshevt(0, x) == 1
assert chebyshevt(1, x) == x
assert chebyshevt(2, x) == 2*x**2 - 1
assert chebyshevt(3, x) == 4*x**3 - 3*x
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert chebyshevt(n, z) == 0
raises(ValueError, lambda: chebyshevt_root(n, n))
for n in range(1, 4):
for k in range(n):
z = chebyshevu_root(n, k)
assert chebyshevu(n, z) == 0
raises(ValueError, lambda: chebyshevu_root(n, n))
n = Symbol("n")
X = chebyshevt(n, x)
assert isinstance(X, chebyshevt)
assert chebyshevt(n, -x) == (-1)**n*chebyshevt(n, x)
assert chebyshevt(-n, x) == chebyshevt(n, x)
assert chebyshevt(n, 0) == cos(pi*n/2)
assert chebyshevt(n, 1) == 1
assert conjugate(chebyshevt(n, x)) == chebyshevt(n, conjugate(x))
assert diff(chebyshevt(n, x), x) == n*chebyshevu(n - 1, x)
X = chebyshevu(n, x)
assert isinstance(X, chebyshevu)
assert chebyshevu(n, -x) == (-1)**n*chebyshevu(n, x)
assert chebyshevu(-n, x) == -chebyshevu(n - 2, x)
assert chebyshevu(n, 0) == cos(pi*n/2)
assert chebyshevu(n, 1) == n + 1
assert conjugate(chebyshevu(n, x)) == chebyshevu(n, conjugate(x))
assert diff(chebyshevu(n, x), x) == \
(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
def test_chebyshev():
assert chebyshevt(0, x) == 1
assert chebyshevt(1, x) == x
assert chebyshevt(2, x) == 2 * x**2 - 1
assert chebyshevt(3, x) == 4 * x**3 - 3 * x
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert chebyshevt(n, z) == 0
raises(ValueError, lambda: chebyshevt_root(n, n))
for n in range(1, 4):
for k in range(n):
z = chebyshevu_root(n, k)
assert chebyshevu(n, z) == 0
raises(ValueError, lambda: chebyshevu_root(n, n))
n = Symbol("n")
X = chebyshevt(n, x)
assert isinstance(X, chebyshevt)
assert chebyshevt(n, -x) == (-1)**n * chebyshevt(n, x)
assert chebyshevt(-n, x) == chebyshevt(n, x)
assert chebyshevt(n, 0) == cos(pi * n / 2)
assert chebyshevt(n, 1) == 1
assert conjugate(chebyshevt(n, x)) == chebyshevt(n, conjugate(x))
assert diff(chebyshevt(n, x), x) == n * chebyshevu(n - 1, x)
X = chebyshevu(n, x)
assert isinstance(X, chebyshevu)
assert chebyshevu(n, -x) == (-1)**n * chebyshevu(n, x)
assert chebyshevu(-n, x) == -chebyshevu(n - 2, x)
assert chebyshevu(n, 0) == cos(pi * n / 2)
assert chebyshevu(n, 1) == n + 1
assert conjugate(chebyshevu(n, x)) == chebyshevu(n, conjugate(x))
assert diff(chebyshevu(n, x), x) == \
(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
def test_chebyshev():
assert chebyshevt(0, x) == 1
assert chebyshevt(1, x) == x
assert chebyshevt(2, x) == 2 * x**2 - 1
assert chebyshevt(3, x) == 4 * x**3 - 3 * x
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert chebyshevt(n, z) == 0
for n in range(1, 4):
for k in range(n):
z = chebyshevu_root(n, k)
assert chebyshevu(n, z) == 0
def test_chebyshev():
assert chebyshevt(0, x) == 1
assert chebyshevt(1, x) == x
assert chebyshevt(2, x) == 2*x**2-1
assert chebyshevt(3, x) == 4*x**3-3*x
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert simplify(chebyshevt(n, z)) == 0
for n in range(1, 4):
for k in range(n):
z = chebyshevu_root(n, k)
assert simplify(chebyshevu(n, z)) == 0