def test_gegenbauer_poly():
raises(ValueError, lambda: gegenbauer_poly(-1, a, x))
assert gegenbauer_poly(1, a, x, polys=True) == Poly(2*a*x, x, domain='ZZ(a)')
assert gegenbauer_poly(0, a, x) == 1
assert gegenbauer_poly(1, a, x) == 2*a*x
assert gegenbauer_poly(2, a, x) == -a + x**2*(2*a**2 + 2*a)
assert gegenbauer_poly(3, a, x) == x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
assert gegenbauer_poly(1, S.Half).dummy_eq(x)
assert gegenbauer_poly(1, a, polys=True) == Poly(2*a*x, x, domain='ZZ(a)')
def test_gegenbauer_poly():
raises(ValueError, lambda: gegenbauer_poly(-1, a, x))
assert gegenbauer_poly(
1, a, x, polys=True) == Poly(2*a*x, x, domain='ZZ(a)')
assert gegenbauer_poly(0, a, x) == 1
assert gegenbauer_poly(1, a, x) == 2*a*x
assert gegenbauer_poly(2, a, x) == -a + x**2*(2*a**2 + 2*a)
assert gegenbauer_poly(
3, a, x) == x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
assert gegenbauer_poly(1, S.Half).dummy_eq(x)
assert gegenbauer_poly(1, a, polys=True) == Poly(2*a*x, x, domain='ZZ(a)')
def eval(cls, n, a, x):
# For negative n the polynomials vanish
# See http://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/
if n.is_negative:
return S.Zero
# Some special values for fixed a
if a == S.Half:
return legendre(n, x)
elif a == S.One:
return chebyshevu(n, x)
elif a == S.NegativeOne:
return S.Zero
if not n.is_Number:
# Handle this before the general sign extraction rule
if x == S.NegativeOne:
if (re(a) > S.Half) == True:
return S.ComplexInfinity
else:
return (
cos(S.Pi * (a + n))
* sec(S.Pi * a)
* gamma(2 * a + n)
/ (gamma(2 * a) * gamma(n + 1))
)
# Symbolic result C^a_n(x)
# C^a_n(-x) ---> (-1)**n * C^a_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne ** n * gegenbauer(n, a, -x)
# We can evaluate for some special values of x
if x.is_zero:
return (
2 ** n
* sqrt(S.Pi)
* gamma(a + S.Half * n)
/ (gamma((1 - n) / 2) * gamma(n + 1) * gamma(a))
)
if x == S.One:
return gamma(2 * a + n) / (gamma(2 * a) * gamma(n + 1))
elif x is S.Infinity:
if n.is_positive:
return RisingFactorial(a, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return gegenbauer_poly(n, a, x)
def eval(cls, n, a, x):
# For negative n the polynomials vanish
# See http://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/
if n.is_negative:
return S.Zero
# Some special values for fixed a
if a == S.Half:
return legendre(n, x)
elif a == S.One:
return chebyshevu(n, x)
elif a == S.NegativeOne:
return S.Zero
if not n.is_Number:
# Handle this before the general sign extraction rule
if x == S.NegativeOne:
if (C.re(a) > S.Half) == True:
return S.ComplexInfinity
else:
# No sec function available yet
# return (C.cos(S.Pi*(a+n)) * C.sec(S.Pi*a) * C.gamma(2*a+n) /
# (C.gamma(2*a) * C.gamma(n+1)))
return None
# Symbolic result C^a_n(x)
# C^a_n(-x) ---> (-1)**n * C^a_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne ** n * gegenbauer(n, a, -x)
# We can evaluate for some special values of x
if x == S.Zero:
return (
2 ** n * sqrt(S.Pi) * C.gamma(a + S.Half * n) / (C.gamma((1 - n) / 2) * C.gamma(n + 1) * C.gamma(a))
)
if x == S.One:
return C.gamma(2 * a + n) / (C.gamma(2 * a) * C.gamma(n + 1))
elif x == S.Infinity:
if n.is_positive:
return C.RisingFactorial(a, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return gegenbauer_poly(n, a, x)
