def legendre_poly(n, x=None, **args):
"""Generates Legendre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Legendre polynomial of degree %s" % n)
poly = DMP(dup_legendre(int(n), QQ), QQ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def chebyshevt_poly(n, x=None, **args):
"""Generates Chebyshev polynomial of the first kind of degree `n` in `x`. """
if n < 0:
raise ValueError(
"can't generate 1st kind Chebyshev polynomial of degree %s" % n)
poly = DMP(dup_chebyshevt(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def jacobi_poly(n, a, b, x=None, **args):
"""Generates Jacobi polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Jacobi polynomial of degree %s" % n)
K, v = construct_domain([a, b], field=True)
poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def cyclotomic_poly(n, x=None, **args):
"""Generates cyclotomic polynomial of order `n` in `x`. """
if n <= 0:
raise ValueError("can't generate cyclotomic polynomial of order %s" %
n)
poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def gegenbauer_poly(n, a, x=None, **args):
"""Generates Gegenbauer polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Gegenbauer polynomial of degree %s" %
n)
K, a = construct_domain(a, field=True)
poly = DMP(dup_gegenbauer(int(n), a, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def spherical_bessel_fn(n, x=None, **args):
"""
Coefficients for the spherical Bessel functions.
Those are only needed in the jn() function.
The coefficients are calculated from:
fn(0, z) = 1/z
fn(1, z) = 1/z**2
fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)
Examples
========
>>> from diofant import Symbol
>>> z = Symbol("z")
>>> spherical_bessel_fn(1, z)
z**(-2)
>>> spherical_bessel_fn(2, z)
-1/z + 3/z**3
>>> spherical_bessel_fn(3, z)
-6/z**2 + 15/z**4
>>> spherical_bessel_fn(4, z)
1/z - 45/z**3 + 105/z**5
"""
if n < 0:
dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
else:
dup = dup_spherical_bessel_fn(int(n), ZZ)
poly = DMP(dup, ZZ)
if x is not None:
poly = Poly.new(poly, 1 / x)
else:
poly = PurePoly.new(poly, 1 / Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def laguerre_poly(n, x=None, alpha=None, **args):
"""Generates Laguerre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Laguerre polynomial of degree %s" % n)
if alpha is not None:
K, alpha = construct_domain(alpha,
field=True) # XXX: ground_field=True
else:
K, alpha = QQ, QQ(0)
poly = DMP(dup_laguerre(int(n), alpha, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly