def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + C.cos(im)**2
return (sinh(re)*cosh(re) + \
S.ImaginaryUnit*C.sin(im)*C.cos(im))/denom
def solve_ODE_second_order(eq, f):
"""
solves many kinds of second order odes, different methods are used
depending on the form of the given equation. Now the constanst
coefficients case and a special case are implemented.
"""
x = f.args[0]
f = f.func
#constant coefficients case: af''(x)+bf'(x)+cf(x)=0
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
c = Wild('c', exclude=[x])
r = eq.match(a * f(x).diff(x, x) + c * f(x))
if r:
return Symbol("C1") * C.sin(sqrt(
r[c] / r[a]) * x) + Symbol("C2") * C.cos(sqrt(r[c] / r[a]) * x)
r = eq.match(a * f(x).diff(x, x) + b * diff(f(x), x) + c * f(x))
if r:
r1 = solve(r[a] * x**2 + r[b] * x + r[c], x)
if r1[0].is_real:
if len(r1) == 1:
return (Symbol("C1") + Symbol("C2") * x) * exp(r1[0] * x)
else:
return Symbol("C1") * exp(r1[0] * x) + Symbol("C2") * exp(
r1[1] * x)
else:
r2 = abs((r1[0] - r1[1]) / (2 * S.ImaginaryUnit))
return (Symbol("C2") * C.cos(r2 * x) +
Symbol("C1") * C.sin(r2 * x)) * exp(
(r1[0] + r1[1]) * x / 2)
#other cases of the second order odes will be implemented here
#special equations, that we know how to solve
t = x * C.exp(f(x))
tt = a * t.diff(x, x) / t
r = eq.match(tt.expand())
if r:
return -solve_ODE_1(f(x), x)
t = x * C.exp(-f(x))
tt = a * t.diff(x, x) / t
r = eq.match(tt.expand())
if r:
#check, that we've rewritten the equation correctly:
#assert ( r[a]*t.diff(x,2)/t ) == eq.subs(f, t)
return solve_ODE_1(f(x), x)
neq = eq * C.exp(f(x)) / C.exp(-f(x))
r = neq.match(tt.expand())
if r:
#check, that we've rewritten the equation correctly:
#assert ( t.diff(x,2)*r[a]/t ).expand() == eq
return solve_ODE_1(f(x), x)
raise NotImplementedError("solve_ODE_second_order: cannot solve " +
str(eq))
def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + C.cos(im)**2
return (sinh(re)*cosh(re) + \
S.ImaginaryUnit*C.sin(im)*C.cos(im))/denom
def as_real_imag(self, deep=True, **hints):
# TODO: Handle deep and hints
n, m, theta, phi = self.args
re = (sqrt((2*n + 1)/(4*pi) * C.factorial(n - m)/C.factorial(n + m)) *
C.cos(m*phi) * assoc_legendre(n, m, C.cos(theta)))
im = (sqrt((2*n + 1)/(4*pi) * C.factorial(n - m)/C.factorial(n + m)) *
C.sin(m*phi) * assoc_legendre(n, m, C.cos(theta)))
return (re, im)
def solve_ODE_second_order(eq, f):
"""
solves many kinds of second order odes, different methods are used
depending on the form of the given equation. So far the constants
coefficients case and a special case are implemented.
"""
x = f.args[0]
f = f.func
#constant coefficients case: af''(x)+bf'(x)+cf(x)=0
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
c = Wild('c', exclude=[x])
r = eq.match(a*f(x).diff(x,x) + c*f(x))
if r:
return Symbol("C1")*C.sin(sqrt(r[c]/r[a])*x)+Symbol("C2")*C.cos(sqrt(r[c]/r[a])*x)
r = eq.match(a*f(x).diff(x,x) + b*diff(f(x),x) + c*f(x))
if r:
r1 = solve(r[a]*x**2 + r[b]*x + r[c], x)
if r1[0].is_real:
if len(r1) == 1:
return (Symbol("C1") + Symbol("C2")*x)*exp(r1[0]*x)
else:
return Symbol("C1")*exp(r1[0]*x) + Symbol("C2")*exp(r1[1]*x)
else:
r2 = abs((r1[0] - r1[1])/(2*S.ImaginaryUnit))
return (Symbol("C2")*C.cos(r2*x) + Symbol("C1")*C.sin(r2*x))*exp((r1[0] + r1[1])*x/2)
#other cases of the second order odes will be implemented here
#special equations, that we know how to solve
a = Wild('a')
t = x*exp(f(x))
tt = a*t.diff(x, x)/t
r = eq.match(tt.expand())
if r:
return -solve_ODE_1(f(x), x)
t = x*exp(-f(x))
tt = a*t.diff(x, x)/t
r = eq.match(tt.expand())
if r:
#check, that we've rewritten the equation correctly:
#assert ( r[a]*t.diff(x,2)/t ) == eq.subs(f, t)
return solve_ODE_1(f(x), x)
neq = eq*exp(f(x))/exp(-f(x))
r = neq.match(tt.expand())
if r:
#check, that we've rewritten the equation correctly:
#assert ( t.diff(x,2)*r[a]/t ).expand() == eq
return solve_ODE_1(f(x), x)
raise NotImplementedError("solve_ODE_second_order: cannot solve " + str(eq))
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints["complex"] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re) ** 2 + C.cos(im) ** 2
return (sinh(re) * cosh(re) / denom, C.sin(im) * C.cos(im) / denom)
def _eval_expand_complex(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return self.expand(deep, **hints)
else:
return self
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + C.cos(im)**2
return (sinh(re)*cosh(re) + \
S.ImaginaryUnit*C.sin(im)*C.cos(im))/denom
def _eval_expand_complex(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return self.expand(deep, **hints)
else:
return self
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + C.cos(im)**2
return (sinh(re)*cosh(re) + \
S.ImaginaryUnit*C.sin(im)*C.cos(im))/denom
def _eval_expand_complex(self, deep=True, **hints):
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = C.cos(im), C.sin(im)
return exp(re) * cos + S.ImaginaryUnit * exp(re) * sin
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.One
elif arg.is_negative:
return cls(-arg)
else:
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cos(i_coeff)
else:
coeff, terms = arg.as_coeff_terms()
if coeff.is_negative:
return cls(-arg)
if isinstance(arg, asinh):
return sqrt(1 + arg.args[0]**2)
if isinstance(arg, acosh):
return arg.args[0]
if isinstance(arg, atanh):
return 1 / sqrt(1 - arg.args[0]**2)
def apothem(self):
"""
Returns the apothem/inradius of the regular polygon (i.e., the
radius of the inscribed circle).
"""
n = self.__getitem__(2)
return self.radius * C.cos(S.Pi/n)
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result U_n(x)
# U_n(-x) ---> (-1)**n * U_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * chebyshevu(n, -x)
# U_{-n}(x) ---> -U_{n-2}(x)
if n.could_extract_minus_sign():
if n == S.NegativeOne:
return S.Zero
else:
return -chebyshevu(-n - 2, x)
# We can evaluate for some special values of x
if x == S.Zero:
return C.cos(S.Half * S.Pi * n)
if x == S.One:
return S.One + n
elif x == S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
# U_{-n}(x) ---> -U_{n-2}(x)
if n == S.NegativeOne:
return S.Zero
else:
return -cls._eval_at_order(-n - 2, x)
else:
return cls._eval_at_order(n, x)
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.One
elif arg.is_negative:
return cls(-arg)
else:
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cos(i_coeff)
else:
coeff, terms = arg.as_coeff_terms()
if coeff.is_negative:
return cls(-arg)
if arg.func == asinh:
return sqrt(1 + arg.args[0] ** 2)
if arg.func == acosh:
return arg.args[0]
if arg.func == atanh:
return 1 / sqrt(1 - arg.args[0] ** 2)
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result U_n(x)
# U_n(-x) ---> (-1)**n * U_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * chebyshevu(n, -x)
# U_{-n}(x) ---> -U_{n-2}(x)
if n.could_extract_minus_sign():
if n == S.NegativeOne:
return S.Zero
else:
return -chebyshevu(-n - 2, x)
# We can evaluate for some special values of x
if x == S.Zero:
return C.cos(S.Half * S.Pi * n)
if x == S.One:
return S.One + n
elif x == S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
# U_{-n}(x) ---> -U_{n-2}(x)
if n == S.NegativeOne:
return S.Zero
else:
return -cls._eval_at_order(-n - 2, x)
else:
return cls._eval_at_order(n, x)
def _eval_expand_complex(self, deep=True, **hints):
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = C.cos(im), C.sin(im)
return exp(re) * cos + S.ImaginaryUnit * exp(re) * sin
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
rv = (sqrt(
(2 * n + 1) / (4 * pi) * C.factorial(n - m) / C.factorial(n + m)) *
C.exp(I * m * phi) * assoc_legendre(n, m, C.cos(theta)))
# We can do this because of the range of theta
return rv.subs(sqrt(-cos(theta)**2 + 1), sin(theta))
def vertices(self):
points = []
c, r, n = self
v = 2*S.Pi/n
for k in xrange(0, n):
points.append( Point(c[0] + r*C.cos(k*v), c[1] + r*C.sin(k*v)) )
return points
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
rv = (
sqrt((2 * n + 1) / (4 * pi) * C.factorial(n - m) / C.factorial(n + m))
* C.exp(I * m * phi)
* assoc_legendre(n, m, C.cos(theta))
)
# We can do this because of the range of theta
return rv.subs(sqrt(-cos(theta) ** 2 + 1), sin(theta))
def arbitrary_point(self, parameter_name='t'):
"""Returns a symbolic point that is on the ellipse."""
t = C.Symbol(parameter_name, real=True)
return Point(self.center[0] + self.hradius * C.cos(t),
self.center[1] + self.vradius * C.sin(t))
def eval(cls, n, k):
if not 0 <= k < n:
raise ValueError("must have 0 <= k < n")
return C.cos(S.Pi*(k+1)/(n+1))
def eval(cls, n, k):
if not 0 <= k < n:
raise ValueError("must have 0 <= k < n")
return C.cos(S.Pi * (k + 1) / (n + 1))
def canonize(cls, n, k):
if not 0 <= k < n:
raise ValueError("must have 0 <= k < n")
return C.cos(S.Pi * (2 * k + 1) / (2 * n))
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
return (sqrt((2*n + 1)/(4*pi) * C.factorial(n - m)/C.factorial(n + m)) *
C.exp(I*m*phi) * assoc_legendre(n, m, C.cos(theta)))
def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
return sinh(re)*C.cos(im) + cosh(re)*C.sin(im)*S.ImaginaryUnit
def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
return sinh(re) * C.cos(im) + cosh(re) * C.sin(im) * S.ImaginaryUnit
def eval(cls, n, k):
if not ((0 <= k) and (k < n)):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return C.cos(S.Pi*(2*k + 1)/(2*n))
def eval(cls, n, k):
if not ((0 <= k) and (k < n)):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return C.cos(S.Pi * (2 * k + 1) / (2 * n))
def _eval_rewrite_as_cos(self, arg):
I = S.ImaginaryUnit
return C.cos(I*arg) + I*C.cos(I*arg+S.Pi/2)
def _eval_expand_complex(self, *args):
re, im = self.args[0].as_real_imag()
cos, sin = C.cos(im), C.sin(im)
return exp(re) * cos + S.ImaginaryUnit * exp(re) * sin
def eval(cls, n, k):
if not ((0 <= k) is (k < n) is True):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return C.cos(S.Pi*(k + 1)/(n + 1))
def eval(cls, n, k):
if not ((0 <= k) is (k < n) is True):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return C.cos(S.Pi * (k + 1) / (n + 1))
def canonize(cls, n, k):
if not 0 <= k < n:
raise ValueError("must have 0 <= k < n")
return C.cos(S.Pi*(2*k+1)/(2*n))
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
return (sqrt(
(2 * n + 1) / (4 * pi) * C.factorial(n - m) / C.factorial(n + m)) *
C.exp(I * m * phi) * assoc_legendre(n, m, C.cos(theta)))
def _eval_expand_complex(self, *args):
re, im = self.args[0].as_real_imag()
cos, sin = C.cos(im), C.sin(im)
return exp(re) * cos + S.ImaginaryUnit * exp(re) * sin
def arbitrary_point(self, parameter_name='t'):
"""Returns a symbolic point that is on the ellipse."""
t = C.Symbol(parameter_name, real=True)
return Point(
self.center[0] + self.hradius*C.cos(t),
self.center[1] + self.vradius*C.sin(t))
def _eval_rewrite_as_cos(self, arg):
I = S.ImaginaryUnit
return C.cos(I*arg) + I*C.cos(I*arg+S.Pi/2)
