def test_ray_derivative(self):
# ray from x=-1 to infinity to the left
gamma = ComplexRay(-1)
self.assertAlmostEqual(gamma.derivative(0), 1)
self.assertAlmostEqual(gamma.derivative(0.5), 4)
self.assertAlmostEqual(gamma.derivative(0.75), 16)
self.assertAlmostEqual(gamma.derivative(1), Infinity)
def test_equality(self):
gamma0 = ComplexLine(-1, 0)
gamma1 = ComplexLine(-1, 0)
self.assertEqual(gamma0, gamma1)
gamma0 = ComplexArc(1, 0, 0, pi)
gamma1 = ComplexArc(1, 0, 0, pi)
self.assertEqual(gamma0, gamma1)
gamma0 = ComplexRay(-1)
gamma1 = ComplexRay(-1)
self.assertEqual(gamma0, gamma1)
def test_ray(self):
# ray from x=-1 to infinity to the left
gamma = ComplexRay(-1)
self.assertAlmostEqual(gamma(0), -1)
self.assertAlmostEqual(gamma(0.5), -2)
self.assertAlmostEqual(gamma(0.75), -4)
self.assertEqual(gamma(1), Infinity)
def test_ray_derivative(self):
# ray from x=-1 to infinity to the left
gamma = ComplexRay(-1)
self.assertAlmostEqual(gamma.derivative(0), 1)
self.assertAlmostEqual(gamma.derivative(0.5), 4)
self.assertAlmostEqual(gamma.derivative(0.75), 16)
self.assertAlmostEqual(gamma.derivative(1), Infinity)
def test_rays(self):
# test that analytic continuation to places at infinity work
gammax = ComplexRay(-9)
y0 = [-3.j, 3.j]
gamma = RiemannSurfacePathPuiseux(self.X1, gammax, y0)
y = gamma.get_y(0)
self.assertAlmostEqual(y[0], -3.j)
self.assertAlmostEqual(y[1], 3.j)
# note: the infinity behavior may change in the future
y = gamma.get_y(1)
self.assertTrue(numpy.isnan(y[0]))
self.assertTrue(numpy.isnan(y[1]))
def _path_to_infinite_place(self, P):
r"""Returns a path to a place at an infintiy of the surface.
A place at infinity is one where the `x`-projection of the place is the
point `x = \infty` of the complex Riemann sphere. An infinite place is
a type of discirminant place.
Parameters
----------
P : Place
The target infinite place.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the place at infinity.
"""
# determine a place Q from where we can reach the target place P. first,
# pick an appropriate x-point over which a Q is chosen
x0 = self.base_point
if numpy.real(x0) < 0:
xa = 5 * x0 # move away from the origin
else:
xa = -5 # arbitrary choice away from the origin
# next, determine an appropriate y-part from where we can reach the
# place at infinity
p = P.puiseux_series
center, coefficient, ramification_index = p.xdata
ta = CC(xa / coefficient).nth_root(abs(ramification_index))
ta = ta if ramification_index > 0 else 1 / ta
p.extend_to_t(ta)
ya = complex(p.eval_y(ta))
# construct the place Q and compute the path going from P0 to Q
Q = self.riemann_surface(xa, ya)
gamma_P0_to_Q = self.path_to_place(Q)
# construct the path going from Q to P
xend = complex(gamma_P0_to_Q.get_x(1.0))
yend = array(gamma_P0_to_Q.get_y(1.0), dtype=complex)
gamma_x = ComplexRay(xend)
gamma_Q_to_P = RiemannSurfacePathPuiseux(self.riemann_surface, gamma_x,
yend)
gamma = gamma_P0_to_Q + gamma_Q_to_P
return gamma
def test_ray(self):
gamma = ComplexRay(1)
self.assertEqual(gamma.x0, 1)