def _eval_expand_trig(self, **hints):
arg = self.args[0]
x = None
if arg.is_Add:
from sympy import symmetric_poly
n = len(arg.args)
TX = []
for x in arg.args:
tx = tan(x, evaluate=False)._eval_expand_trig()
TX.append(tx)
Yg = numbered_symbols('Y')
Y = [Yg.next() for i in xrange(n)]
p = [0, 0]
for i in xrange(n + 1):
p[1 - i % 2] += symmetric_poly(i, Y) * (-1)**((i % 4) // 2)
return (p[0] / p[1]).subs(zip(Y, TX))
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = C.Symbol('dummy', real=True)
P = ((1 + I * z)**coeff).expand()
return (C.im(P) / C.re(P)).subs([(z, tan(terms))])
return tan(arg)
def _eval_expand_trig(self, **hints):
arg = self.args[0]
x = None
if arg.is_Add:
from sympy import symmetric_poly
n = len(arg.args)
TX = []
for x in arg.args:
tx = tan(x, evaluate=False)._eval_expand_trig()
TX.append(tx)
Yg = numbered_symbols('Y')
Y = [ Yg.next() for i in xrange(n) ]
p = [0,0]
for i in xrange(n+1):
p[1-i%2] += symmetric_poly(i,Y)*(-1)**((i%4)//2)
return (p[0]/p[1]).subs(zip(Y,TX))
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = C.Symbol('dummy',real=True)
P = ((1+I*z)**coeff).expand()
return (C.im(P)/C.re(P)).subs([(z,tan(terms))])
return tan(arg)
def _eval_expand_trig(self, **hints):
arg = self.args[0]
x = None
if arg.is_Add:
from sympy import symmetric_poly
n = len(arg.args)
CX = []
for x in arg.args:
cx = cot(x, evaluate=False)._eval_expand_trig()
CX.append(cx)
Yg = numbered_symbols("Y")
Y = [Yg.next() for i in xrange(n)]
p = [0, 0]
for i in xrange(n, -1, -1):
p[(n - i) % 2] += symmetric_poly(i, Y) * (-1) ** (((n - i) % 4) // 2)
return (p[0] / p[1]).subs(zip(Y, CX))
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = C.Symbol("dummy", real=True)
P = ((z + I) ** coeff).expand()
return (C.re(P) / C.im(P)).subs([(z, cot(terms))])
return cot(arg)
def as_real_imag(self, deep=True, **hints):
other = []
coeff = S(1)
for a in self.args:
if a.is_real:
coeff *= a
else:
other.append(a)
m = Mul(*other)
if hints.get('ignore') == m:
return None
else:
return (coeff*C.re(m), coeff*C.im(m))
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)
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) is 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)
def as_real_imag(self, deep=True, **hints):
other = []
coeff = S(1)
for a in self.args:
if a.is_real:
coeff *= a
elif a.is_commutative:
# search for complex conjugate pairs:
for i, x in enumerate(other):
if x == a.conjugate():
coeff *= C.Abs(x)**2
del other[i]
break
else:
other.append(a)
else:
other.append(a)
m = Mul(*other)
if hints.get('ignore') == m:
return None
else:
return (coeff*C.re(m), coeff*C.im(m))
def as_real_imag(self, deep=True, **hints):
other = []
coeff = S(1)
for a in self.args:
if a.is_real:
coeff *= a
elif a.is_commutative:
# search for complex conjugate pairs:
for i, x in enumerate(other):
if x == a.conjugate():
coeff *= C.Abs(x)**2
del other[i]
break
else:
other.append(a)
else:
other.append(a)
m = Mul(*other)
if hints.get('ignore') == m:
return None
else:
return (coeff * C.re(m), coeff * C.im(m))
