def eval(cls, nu, z):
if z.is_zero:
if nu.is_zero:
return S.One
elif (nu.is_integer and nu.is_zero is False) or re(nu).is_positive:
return S.Zero
elif re(nu).is_negative and not (nu.is_integer is True):
return S.ComplexInfinity
elif nu.is_imaginary:
return S.NaN
if im(z) is S.Infinity or im(z) is S.NegativeInfinity:
return S.Zero
if z.could_extract_minus_sign():
return (z)**nu * (-z)**(-nu) * besseli(nu, -z)
if nu.is_integer:
if nu.could_extract_minus_sign():
return besseli(-nu, z)
newz = z.extract_multiplicatively(I)
if newz: # NOTE we don't want to change the function if z==0
return I**(-nu) * besselj(nu, -newz)
# branch handling:
from diofant import unpolarify, exp
if nu.is_integer:
newz = unpolarify(z)
if newz != z:
return besseli(nu, newz)
else:
newz, n = z.extract_branch_factor()
if n != 0:
return exp(2 * n * pi * nu * I) * besseli(nu, newz)
nnu = unpolarify(nu)
if nu != nnu:
return besseli(nnu, z)
def eval(cls, nu, z):
if z.is_zero:
if nu.is_zero:
return S.Infinity
elif re(nu).is_zero is False:
return S.ComplexInfinity
elif re(nu).is_zero:
return S.NaN
if im(z) is S.Infinity or im(z) is S.NegativeInfinity:
return S.Zero
if nu.is_integer:
if nu.could_extract_minus_sign():
return besselk(-nu, z)
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(z, Rational(3, 2))
if re(z).is_positive:
return ot * sqrt(z) * (besseli(-ot, tt * a) - besseli(ot, tt * a))
else:
return ot * (Pow(a, ot) * besseli(-ot, tt * a) -
z * Pow(a, -ot) * besseli(ot, tt * a))
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = tt * Pow(z, Rational(3, 2))
if re(z).is_positive:
return z / sqrt(3) * (besseli(-tt, a) + besseli(tt, a))
else:
a = Pow(z, Rational(3, 2))
b = Pow(a, tt)
c = Pow(a, -tt)
return sqrt(ot) * (b * besseli(-tt, tt * a) +
z**2 * c * besseli(tt, tt * a))
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = tt * Pow(z, Rational(3, 2))
if re(z).is_positive:
return z / 3 * (besseli(tt, a) - besseli(-tt, a))
else:
a = Pow(z, Rational(3, 2))
b = Pow(a, tt)
c = Pow(a, -tt)
return ot * (z**2 * c * besseli(tt, tt * a) -
b * besseli(-ot, tt * a))
def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(z, Rational(3, 2))
if re(z).is_positive:
return sqrt(z) / sqrt(3) * (besseli(-ot, tt * a) +
besseli(ot, tt * a))
else:
b = Pow(a, ot)
c = Pow(a, -ot)
return sqrt(ot) * (b * besseli(-ot, tt * a) +
z * c * besseli(ot, tt * 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) is S.true:
return S.ComplexInfinity
else:
# No sec function available yet
# return (cos(S.Pi*(a+n)) * sec(S.Pi*a) * gamma(2*a+n) /
# (gamma(2*a) * gamma(n+1)))
return
# 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) * 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 == 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_rewrite_as_besselj(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(-z, Rational(3, 2))
if re(z).is_negative:
return ot * sqrt(-z) * (besselj(-ot, tt * a) + besselj(ot, tt * a))
def _eval_rewrite_as_besselj(self, z):
tt = Rational(2, 3)
a = tt * Pow(-z, Rational(3, 2))
if re(z).is_negative:
return -z / sqrt(3) * (besselj(-tt, a) + besselj(tt, a))
def _eval_rewrite_as_besselj(self, z):
tt = Rational(2, 3)
a = Pow(-z, Rational(3, 2))
if re(z).is_negative:
return z / 3 * (besselj(-tt, tt * a) - besselj(tt, tt * a))