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 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 test_analytic_continuation_X1(self):
gammax = ComplexLine(1, 0)
y0 = [-1, 1]
gamma = RiemannSurfacePathPuiseux(self.X1, gammax, y0)
y = gamma.get_y(0)
self.assertAlmostEqual(y[0], -1)
self.assertAlmostEqual(y[1], 1)
y = gamma.get_y(0.5)
self.assertAlmostEqual(y[0], -sqrt(complex(0.5)))
self.assertAlmostEqual(y[1], sqrt(complex(0.5)))
y = gamma.get_y(0.75)
self.assertAlmostEqual(y[0], -sqrt(complex(0.25)))
self.assertAlmostEqual(y[1], sqrt(complex(0.25)))
y = gamma.get_y(1)
self.assertAlmostEqual(y[0], 0)
self.assertAlmostEqual(y[1], 0)
gammax = ComplexArc(2, 2, 0, pi)
y0 = [-2, 2]
gamma = RiemannSurfacePathPuiseux(self.X1, gammax, y0)
y = gamma.get_y(0)
self.assertAlmostEqual(y[0], -2)
self.assertAlmostEqual(y[1], 2)
y = gamma.get_y(1)
self.assertAlmostEqual(y[0], 0)
self.assertAlmostEqual(y[1], 0)
def test_analytic_continuation_X1(self):
gammax = ComplexLine(1,0)
y0 = [-1,1]
gamma = RiemannSurfacePathPuiseux(self.X1, gammax, y0)
y = gamma.get_y(0)
self.assertAlmostEqual(y[0], -1)
self.assertAlmostEqual(y[1], 1)
y = gamma.get_y(0.5)
self.assertAlmostEqual(y[0], -sqrt(complex(0.5)))
self.assertAlmostEqual(y[1], sqrt(complex(0.5)))
y = gamma.get_y(0.75)
self.assertAlmostEqual(y[0], -sqrt(complex(0.25)))
self.assertAlmostEqual(y[1], sqrt(complex(0.25)))
y = gamma.get_y(1)
self.assertAlmostEqual(y[0], 0)
self.assertAlmostEqual(y[1], 0)
gammax = ComplexArc(2,2,0,pi)
y0 = [-2,2]
gamma = RiemannSurfacePathPuiseux(self.X1, gammax, y0)
y = gamma.get_y(0)
self.assertAlmostEqual(y[0], -2)
self.assertAlmostEqual(y[1], 2)
y = gamma.get_y(1)
self.assertAlmostEqual(y[0], 0)
self.assertAlmostEqual(y[1], 0)
def test_analytic_continuation_X2(self):
S = QQ['t']; t = S.gen()
a,b,c = (t**3 - 1).roots(ring=QQbar, multiplicities=False)
gammax = ComplexLine(1,0)
y0 = [a,b,c]
gamma = RiemannSurfacePathPuiseux(self.X2, gammax, y0)
y = gamma.get_y(0)
self.assertAlmostEqual(y[0], a)
self.assertAlmostEqual(y[1], b)
self.assertAlmostEqual(y[2], c)
scale = (0.5)**(1/3.)
y = gamma.get_y(0.5)
self.assertAlmostEqual(y[0], scale*a)
self.assertAlmostEqual(y[1], scale*b)
self.assertAlmostEqual(y[2], scale*c)
y = gamma.get_y(1)
self.assertAlmostEqual(y[0], 0)
self.assertAlmostEqual(y[1], 0)
self.assertAlmostEqual(y[2], 0)
def test_analytic_continuation_X2(self):
S = QQ['t']
t = S.gen()
a, b, c = (t**3 - 1).roots(ring=QQbar, multiplicities=False)
gammax = ComplexLine(1, 0)
y0 = [a, b, c]
gamma = RiemannSurfacePathPuiseux(self.X2, gammax, y0)
y = gamma.get_y(0)
self.assertAlmostEqual(y[0], a)
self.assertAlmostEqual(y[1], b)
self.assertAlmostEqual(y[2], c)
scale = (0.5)**(1 / 3.)
y = gamma.get_y(0.5)
self.assertAlmostEqual(y[0], scale * a)
self.assertAlmostEqual(y[1], scale * b)
self.assertAlmostEqual(y[2], scale * c)
y = gamma.get_y(1)
self.assertAlmostEqual(y[0], 0)
self.assertAlmostEqual(y[1], 0)
self.assertAlmostEqual(y[2], 0)
