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) * factorial(n - m)/factorial(n + m)) *
cos(m*phi) * assoc_legendre(n, m, cos(theta)))
im = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
sin(m*phi) * assoc_legendre(n, m, cos(theta)))
return re, im
def test_sympyissue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == cbrt((1 + I)**(12*I*f))
assert cbrt((1 + I)**(I*(1 + 7*f))).exp == Rational(1, 3)
r = symbols('r', extended_real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, Rational(5, 2))) != (2*I)**Rational(5, 4)
p = symbols('p', positive=True)
assert cbrt(p**2) == p**Rational(2, 3)
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt((1 - I)/2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**Rational(-1, 2) == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == Rational(1, 2)
assert sqrt(r**Rational(4, 3)) != r**Rational(2, 3)
assert sqrt((p + I)**Rational(4, 3)) == (p + I)**Rational(2, 3)
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**Rational(5, 2)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_sympyissue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == ((1 + I)**(12*I*f))**Rational(1, 3)
assert (((1 + I)**(I*(1 + 7*f)))**Rational(1, 3)).exp == Rational(1, 3)
r = symbols('r', extended_real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, 5*S.Half)) != (2*I)**(5/Integer(4))
p = symbols('p', positive=True)
assert cbrt(p**2) == p**(2/Integer(3))
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt((1 - I)/2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**-S.Half == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == S.Half
assert sqrt(r**(4/Integer(3))) != r**(2/Integer(3))
assert sqrt((p + I)**(4/Integer(3))) == (p + I)**(2/Integer(3))
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**(5*S.Half)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def _eval_expand_func(self, **hints):
from diofant import Sum
n = self.args[0]
m = self.args[1] if len(self.args) == 2 else 1
if m == S.One:
if n.is_Add:
off = n.args[0]
nnew = n - off
if off.is_Integer and off.is_positive:
result = [S.One / (nnew + i)
for i in range(off, 0, -1)] + [harmonic(nnew)]
return Add(*result)
elif off.is_Integer and off.is_negative:
result = [-S.One / (nnew + i)
for i in range(0, off, -1)] + [harmonic(nnew)]
return Add(*result)
if n.is_Rational:
# Expansions for harmonic numbers at general rational arguments (u + p/q)
# Split n as u + p/q with p < q
p, q = n.as_numer_denom()
u = p // q
p = p - u * q
if u.is_nonnegative and p.is_positive and q.is_positive and p < q:
k = Dummy("k")
t1 = q * Sum(1 / (q * k + p), (k, 0, u))
t2 = 2 * Sum(
cos((2 * pi * p * k) / q) * log(sin((pi * k) / q)),
(k, 1, floor((q - 1) / Integer(2))))
t3 = (pi / 2) * cot((pi * p) / q) + log(2 * q)
return t1 + t2 - t3
return self
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 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 test_Max():
from diofant.abc import x, y, z
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
nn_ = Symbol('nn_', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
np = Symbol('np', nonpositive=True)
np_ = Symbol('np_', nonpositive=True)
r = Symbol('r', extended_real=True)
assert Max(5, 4) == 5
# lists
pytest.raises(ValueError, lambda: Max())
assert Max(x, y) == Max(y, x)
assert Max(x, y, z) == Max(z, y, x)
assert Max(x, Max(y, z)) == Max(z, y, x)
assert Max(x, Min(y, oo)) == Max(x, y)
assert Max(n, -oo, n_, p, 2) == Max(p, 2)
assert Max(n, -oo, n_, p) == p
assert Max(2, x, p, n, -oo, S.NegativeInfinity, n_, p, 2) == Max(2, x, p)
assert Max(0, x, 1, y) == Max(1, x, y)
assert Max(r, r + 1, r - 1) == 1 + r
assert Max(1000, 100, -100, x, p, n) == Max(p, x, 1000)
assert Max(cos(x), sin(x)) == Max(sin(x), cos(x))
assert Max(cos(x), sin(x)).subs(x, 1) == sin(1)
assert Max(cos(x), sin(x)).subs(x, Rational(1, 2)) == cos(Rational(1, 2))
pytest.raises(ValueError, lambda: Max(cos(x), sin(x)).subs(x, I))
pytest.raises(ValueError, lambda: Max(I))
pytest.raises(ValueError, lambda: Max(I, x))
pytest.raises(ValueError, lambda: Max(S.ComplexInfinity, 1))
# interesting:
# Max(n, -oo, n_, p, 2) == Max(p, 2)
# True
# Max(n, -oo, n_, p, 1000) == Max(p, 1000)
# False
assert Max(1, x).diff(x) == Heaviside(x - 1)
assert Max(x, 1).diff(x) == Heaviside(x - 1)
assert Max(x**2, 1 + x, 1).diff(x) == \
2*x*Heaviside(x**2 - Max(1, x + 1)) \
+ Heaviside(x - Max(1, x**2) + 1)
a, b = Symbol('a', extended_real=True), Symbol('b', extended_real=True)
# a and b are both real, Max(a, b) should be real
assert Max(a, b).is_extended_real
# issue 7233
e = Max(0, x)
assert e.evalf == e.n
assert e.n().args == (0, x)
def taylor_term(n, x, *previous_terms):
if n < 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 1:
p = previous_terms[-1]
return (3**Rational(1, 3) * x *
Abs(sin(2 * pi *
(n + S.One) / Integer(3))) * factorial(
(n - S.One) / Integer(3)) /
((n + S.One) *
Abs(cos(2 * pi *
(n + S.Half) / Integer(3))) * factorial(
(n - 2) / Integer(3))) * p)
else:
return (S.One / (root(3, 6) * pi) * gamma(
(n + S.One) / Integer(3)) *
Abs(sin(2 * pi * (n + S.One) / Integer(3))) /
factorial(n) * (root(3, 3) * x)**n)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the functions.
See Also
========
diofant.geometry.point.Point
Examples
========
>>> from diofant import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point2D(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(
filldedent('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name))
return Point(self.center.x + self.hradius * cos(t),
self.center.y + self.vradius * sin(t))
def rot_axis1(theta):
"""Returns a rotation matrix for a rotation of theta (in radians) about
the 1-axis.
Examples
========
>>> from diofant import pi
>>> from diofant.matrices import rot_axis1
A rotation of pi/3 (60 degrees):
>>> theta = pi/3
>>> rot_axis1(theta)
Matrix([
[1, 0, 0],
[0, 1/2, sqrt(3)/2],
[0, -sqrt(3)/2, 1/2]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis1(pi/2)
Matrix([
[1, 0, 0],
[0, 0, 1],
[0, -1, 0]])
See Also
========
diofant.matrices.dense.rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
about the 2-axis
diofant.matrices.dense.rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
about the 3-axis
"""
ct = cos(theta)
st = sin(theta)
lil = ((1, 0, 0), (0, ct, st), (0, -st, ct))
return Matrix(lil)
def _minpoly_sin(ex, x):
"""
Returns the minimal polynomial of ``sin(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
n = c.q
q = sympify(n)
if q.is_prime:
# for a = pi*p/q with q odd prime, using chebyshevt
# write sin(q*a) = mp(sin(a))*sin(a);
# the roots of mp(x) are sin(pi*p/q) for p = 1,..., q - 1
a = dup_chebyshevt(n, ZZ)
return Add(*[x**(n - i - 1) * a[i] for i in range(n)])
if c.p == 1:
if q == 9:
return 64 * x**6 - 96 * x**4 + 36 * x**2 - 3
if n % 2 == 1:
# for a = pi*p/q with q odd, use
# sin(q*a) = 0 to see that the minimal polynomial must be
# a factor of dup_chebyshevt(n, ZZ)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i) * a[i] for i in range(n + 1)]
r = Add(*a)
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
expr = ((1 - cos(2 * c * pi)) / 2)**S.Half
res = _minpoly_compose(expr, x, QQ)
return res
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
See Also
========
diofant.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
Notes
-----
A random point may not appear to be on the ellipse, ie, `p in e` may
return False. This is because the coordinates of the point will be
floating point values, and when these values are substituted into the
equation for the ellipse the result may not be zero because of floating
point rounding error.
Examples
========
>>> from diofant import Point, Ellipse, Segment
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point2D(2.1, 1.4)
The random_point method assures that the point will test as being
in the ellipse:
>>> p1 in e1
True
Notes
=====
An arbitrary_point with a random value of t substituted into it may
not test as being on the ellipse because the expression tested that
a point is on the ellipse doesn't simplify to zero and doesn't evaluate
exactly to zero:
>>> from diofant.abc import t
>>> e1.arbitrary_point(t)
Point2D(3*cos(t), 2*sin(t))
>>> p2 = _.subs(t, 0.1)
>>> p2 in e1
False
Note that arbitrary_point routine does not take this approach. A value
for cos(t) and sin(t) (not t) is substituted into the arbitrary point.
There is a small chance that this will give a point that will not
test as being in the ellipse, so the process is repeated (up to 10
times) until a valid point is obtained.
"""
from diofant import sin, cos, Rational
t = _symbol('t')
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
for i in range(10): # should be enough?
# simplify this now or else the Float will turn s into a Float
c = 2 * Rational(rng.random()) - 1
s = sqrt(1 - c**2)
p1 = Point(x.subs(cos(t), c), y.subs(sin(t), s))
if p1 in self:
return p1
raise GeometryError(
'Having problems generating a point in the ellipse.')
def _eval_expand_func(self, **hints):
n, m, theta, phi = self.args
rv = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
exp(I*m*phi) * assoc_legendre(n, m, cos(theta)))
# We can do this because of the range of theta
return rv.subs(sqrt(-cos(theta)**2 + 1), sin(theta))
def _eval_rewrite_as_besselj(self, nu, z):
if nu.is_integer is False:
return csc(
pi * nu) * (cos(pi * nu) * besselj(nu, z) - besselj(-nu, z))
def gauss_chebyshev_u(n, n_digits):
r"""
Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
the second kind.
The Gauss-Chebyshev quadrature of the second kind approximates the integral:
.. math::
\int_{-1}^{1} \sqrt{1-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `U_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{\pi}{n+1} \sin^2 \left(\frac{i}{n+1}\pi\right)
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from diofant import S, pprint
>>> from diofant.integrals.quadrature import gauss_chebyshev_u
>>> x, w = gauss_chebyshev_u(3, 5)
>>> x
[0.70711, 0, -0.70711]
>>> pprint(w)
[0.3927, 0.7854, 0.3927]
>>> x, w = gauss_chebyshev_u(6, 5)
>>> x
[0.90097, 0.62349, 0.22252, -0.22252, -0.62349, -0.90097]
>>> w
[0.084489, 0.27433, 0.42658, 0.42658, 0.27433, 0.084489]
See Also
========
diofant.integrals.quadrature.gauss_legendre
diofant.integrals.quadrature.gauss_laguerre
diofant.integrals.quadrature.gauss_hermite
diofant.integrals.quadrature.gauss_gen_laguerre
diofant.integrals.quadrature.gauss_chebyshev_t
diofant.integrals.quadrature.gauss_jacobi
References
==========
.. [1] http://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev2_rule/chebyshev2_rule.html
"""
x = Dummy("x")
xi = []
w = []
for i in range(1, n + 1):
xi.append((cos(i / (n + S.One) * S.Pi)).n(n_digits))
w.append(
(S.Pi / (n + S.One) * sin(i * S.Pi / (n + S.One))**2).n(n_digits))
return xi, w
def gauss_chebyshev_t(n, n_digits):
r"""
Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
the first kind.
The Gauss-Chebyshev quadrature of the first kind approximates the integral:
.. math::
\int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
The nodes `x_i` of an order `n` quadrature rule are the roots of `T_n`
and the weights `w_i` are given by:
.. math::
w_i = \frac{\pi}{n}
Parameters
==========
n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns
=======
(x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
The points `x_i` and weights `w_i` are returned as ``(x, w)``
tuple of lists.
Examples
========
>>> from diofant import S, pprint
>>> from diofant.integrals.quadrature import gauss_chebyshev_t
>>> x, w = gauss_chebyshev_t(3, 5)
>>> x
[0.86602, 0, -0.86602]
>>> w
[1.0472, 1.0472, 1.0472]
>>> x, w = gauss_chebyshev_t(6, 5)
>>> x
[0.96593, 0.70711, 0.25882, -0.25882, -0.70711, -0.96593]
>>> pprint(w)
[0.5236, 0.5236, 0.5236, 0.5236, 0.5236, 0.5236]
See Also
========
diofant.integrals.quadrature.gauss_legendre
diofant.integrals.quadrature.gauss_laguerre
diofant.integrals.quadrature.gauss_hermite
diofant.integrals.quadrature.gauss_gen_laguerre
diofant.integrals.quadrature.gauss_chebyshev_u
diofant.integrals.quadrature.gauss_jacobi
References
==========
.. [1] http://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
.. [2] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev1_rule/chebyshev1_rule.html
"""
x = Dummy("x")
xi = []
w = []
for i in range(1, n + 1):
xi.append((cos((2 * i - S.One) / (2 * n) * S.Pi)).n(n_digits))
w.append((S.Pi / n).n(n_digits))
return xi, w
def test_Min():
from diofant.abc import x, y, z
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
nn_ = Symbol('nn_', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
np = Symbol('np', nonpositive=True)
np_ = Symbol('np_', nonpositive=True)
assert Min(5, 4) == 4
assert Min(-oo, -oo) == -oo
assert Min(-oo, n) == -oo
assert Min(n, -oo) == -oo
assert Min(-oo, np) == -oo
assert Min(np, -oo) == -oo
assert Min(-oo, 0) == -oo
assert Min(0, -oo) == -oo
assert Min(-oo, nn) == -oo
assert Min(nn, -oo) == -oo
assert Min(-oo, p) == -oo
assert Min(p, -oo) == -oo
assert Min(-oo, oo) == -oo
assert Min(oo, -oo) == -oo
assert Min(n, n) == n
assert Min(n, np) == Min(n, np)
assert Min(np, n) == Min(np, n)
assert Min(n, 0) == n
assert Min(0, n) == n
assert Min(n, nn) == n
assert Min(nn, n) == n
assert Min(n, p) == n
assert Min(p, n) == n
assert Min(n, oo) == n
assert Min(oo, n) == n
assert Min(np, np) == np
assert Min(np, 0) == np
assert Min(0, np) == np
assert Min(np, nn) == np
assert Min(nn, np) == np
assert Min(np, p) == np
assert Min(p, np) == np
assert Min(np, oo) == np
assert Min(oo, np) == np
assert Min(0, 0) == 0
assert Min(0, nn) == 0
assert Min(nn, 0) == 0
assert Min(0, p) == 0
assert Min(p, 0) == 0
assert Min(0, oo) == 0
assert Min(oo, 0) == 0
assert Min(nn, nn) == nn
assert Min(nn, p) == Min(nn, p)
assert Min(p, nn) == Min(p, nn)
assert Min(nn, oo) == nn
assert Min(oo, nn) == nn
assert Min(p, p) == p
assert Min(p, oo) == p
assert Min(oo, p) == p
assert Min(oo, oo) == oo
assert Min(n, n_).func is Min
assert Min(nn, nn_).func is Min
assert Min(np, np_).func is Min
assert Min(p, p_).func is Min
# lists
pytest.raises(ValueError, lambda: Min())
assert Min(x, y) == Min(y, x)
assert Min(x, y, z) == Min(z, y, x)
assert Min(x, Min(y, z)) == Min(z, y, x)
assert Min(x, Max(y, -oo)) == Min(x, y)
assert Min(p, oo, n, p, p, p_) == n
assert Min(p_, n_, p) == n_
assert Min(n, oo, -7, p, p, 2) == Min(n, -7)
assert Min(2, x, p, n, oo, n_, p, 2, -2, -2) == Min(-2, x, n, n_)
assert Min(0, x, 1, y) == Min(0, x, y)
assert Min(1000, 100, -100, x, p, n) == Min(n, x, -100)
assert Min(cos(x), sin(x)) == Min(cos(x), sin(x))
assert Min(cos(x), sin(x)).subs(x, 1) == cos(1)
assert Min(cos(x), sin(x)).subs(x, Rational(1, 2)) == sin(Rational(1, 2))
pytest.raises(ValueError, lambda: Min(cos(x), sin(x)).subs(x, I))
pytest.raises(ValueError, lambda: Min(I))
pytest.raises(ValueError, lambda: Min(I, x))
pytest.raises(ValueError, lambda: Min(S.ComplexInfinity, x))
assert Min(1, x).diff(x) == Heaviside(1 - x)
assert Min(x, 1).diff(x) == Heaviside(1 - x)
assert Min(0, -x, 1 - 2*x).diff(x) == -Heaviside(x + Min(0, -2*x + 1)) \
- 2*Heaviside(2*x + Min(0, -x) - 1)
a, b = Symbol('a', extended_real=True), Symbol('b', extended_real=True)
# a and b are both real, Min(a, b) should be real
assert Min(a, b).is_extended_real
# issue 7619
f = Function('f')
assert Min(1, 2 * Min(f(1), 2)) # doesn't fail
# issue 7233
e = Min(0, x)
assert e.evalf == e.n
assert e.n().args == (0, x)
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 cos(S.Pi*(k + 1)/(n + 1))
def _expand(self, **hints):
n = self.order
z = self.argument
return (-1)**(n + 1) * \
(fn(-n - 1, z) * sin(z) + (-1)**(-n) * fn(n, z) * cos(z))
def _eval_rewrite_as_sqrt(self, n, sym=None):
from diofant.functions import sqrt
if sym is None:
return (S.GoldenRatio**n -
cos(S.Pi * n) / S.GoldenRatio**n) / sqrt(5)
