def test_jacobi():
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(2, a, b, x) == a**2 / 8 - a * b / 4 - a / 8 + b**2 / 8 -
b / 8 + x**2 * (a**2 / 8 + a * b / 4 + 7 * a / 8 + b**2 / 8 +
7 * b / 8 + Rational(3, 2)) + x *
(a**2 / 4 + 3 * a / 4 - b**2 / 4 - 3 * b / 4) - S.Half)
assert jacobi(n, a, a, x) == RisingFactorial(a + 1, n) * gegenbauer(
n, a + Rational(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(Rational(
3, 2), n) * chebyshevu(n, x) / factorial(n + 1)
assert jacobi(n, -S.Half, -S.Half, x) == RisingFactorial(
Rational(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 + Rational(1, 2))*jacobi(n - 1, a + 1, b + 1, x)
# XXX see issue sympy/sympy#5539
assert str(jacobi(n, a, b, x).diff(a)) == \
("Sum((jacobi(n, a, b, x) + (a + b + 2*_k + 1)*RisingFactorial(b + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)))/(a + b + n + _k + 1), (_k, 0, n - 1))")
assert str(jacobi(n, a, b, x).diff(b)) == \
("Sum(((-1)**(n - _k)*(a + b + 2*_k + 1)*RisingFactorial(a + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)) + jacobi(n, a, b, x))/(a + b + n + "
"_k + 1), (_k, 0, n - 1))")
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))))
pytest.raises(ValueError, lambda: jacobi(-2.1, a, b, x))
pytest.raises(ValueError,
lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
pytest.raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
def test_jacobi():
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(2, a, b, x) == a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 +
x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + Rational(3, 2)) +
x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - Rational(1, 2))
assert jacobi(n, a, a, x) == RisingFactorial(
a + 1, n)*gegenbauer(n, a + Rational(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, Rational(1, 2), Rational(1, 2), x) == RisingFactorial(
Rational(3, 2), n)*chebyshevu(n, x)/factorial(n + 1)
assert jacobi(n, Rational(-1, 2), Rational(-1, 2), x) == RisingFactorial(
Rational(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 jacobi(n, a, b, oo) == jacobi(n, a, b, oo, evaluate=False)
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 + Rational(1, 2))*jacobi(n - 1, a + 1, b + 1, x)
# XXX see issue sympy/sympy#5539
assert str(jacobi(n, a, b, x).diff(a)) == \
("Sum((jacobi(n, a, b, x) + (a + b + 2*_k + 1)*RisingFactorial(b + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)))/(a + b + n + _k + 1), (_k, 0, n - 1))")
assert str(jacobi(n, a, b, x).diff(b)) == \
("Sum(((-1)**(n - _k)*(a + b + 2*_k + 1)*RisingFactorial(a + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)) + jacobi(n, a, b, x))/(a + b + n + "
"_k + 1), (_k, 0, n - 1))")
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))))
pytest.raises(ValueError, lambda: jacobi(-2.1, a, b, x))
pytest.raises(ValueError, lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
pytest.raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
def test_gegenbauer():
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, Rational(1, 2), x) == legendre(n, x)
assert gegenbauer(n, 1, x) == chebyshevu(n, x)
assert gegenbauer(n, -1, x) == 0
assert gegenbauer(n, -2, -1) == gegenbauer(n, -2, -1, evaluate=False)
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 + Rational(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 gegenbauer(n, a, oo) == gegenbauer(n, a, oo, evaluate=False)
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)
pytest.raises(ArgumentIndexError, lambda: gegenbauer(n, a, x).fdiff(4))
# XXX see issue sympy/sympy#5539
assert str(gegenbauer(n, a, x).diff(a)) == \
("Sum((2*(-1)**(n - _k) + 2)*(a + _k)*gegenbauer(_k, a, x)/((n - "
"_k)*(2*a + n + _k)) + (2/(2*a + n + _k) + (2*_k + 2)/((2*a + "
"_k)*(2*a + 2*_k + 1)))*gegenbauer(n, a, x), (_k, 0, n - 1))")
def test_gegenbauer():
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, Rational(1, 2), x) == legendre(n, x)
assert gegenbauer(n, 1, x) == chebyshevu(n, x)
assert gegenbauer(n, -1, x) == 0
assert gegenbauer(n, -2, -1) == gegenbauer(n, -2, -1, evaluate=False)
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 + Rational(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 gegenbauer(n, a, oo) == gegenbauer(n, a, oo, evaluate=False)
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)
pytest.raises(ArgumentIndexError, lambda: gegenbauer(n, a, x).fdiff(4))
# XXX see issue sympy/sympy#5539
assert str(gegenbauer(n, a, x).diff(a)) == \
("Sum((2*(-1)**(n - _k) + 2)*(a + _k)*gegenbauer(_k, a, x)/((n - "
"_k)*(2*a + n + _k)) + (2/(2*a + n + _k) + (2*_k + 2)/((2*a + "
"_k)*(2*a + 2*_k + 1)))*gegenbauer(n, a, x), (_k, 0, n - 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
pytest.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
pytest.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, oo) == oo
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)
pytest.raises(ArgumentIndexError, lambda: chebyshevt(n, x).fdiff(1))
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, oo) == oo
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)
pytest.raises(ArgumentIndexError, lambda: chebyshevu(n, x).fdiff(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
assert chebyshevt(-2, x) == 2*x**2 - 1
for n in range(1, 4):
for k in range(n):
z = chebyshevt_root(n, k)
assert chebyshevt(n, z) == 0
pytest.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
pytest.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, oo) == oo
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)
pytest.raises(ArgumentIndexError, lambda: chebyshevt(n, x).fdiff(1))
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, oo) == oo
assert chebyshevu(n, 0) == cos(pi*n/2)
assert chebyshevu(n, 1) == n + 1
assert chebyshevu(-1, x) == 0
assert chebyshevu(-2, x) == -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)
pytest.raises(ArgumentIndexError, lambda: chebyshevu(n, x).fdiff(1))
