def RQ(delta):
# this is the quotient R(F_0,z)/R(F_0,z(F)) for a generic z
# at distance delta from j. See Lemma 4.2 in [HS2018].
cd = cosh(delta).n(prec=prec)
sd = sinh(delta).n(prec=prec)
return prod(
[cd + (cost * phi[0] + sint * phi[1]) * sd for phi in phis])
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
if (not isinstance(n, Expression) and not isinstance(x, Expression)
and (is_inexact(n) or is_inexact(x))):
coercion_model = get_coercion_model()
n, x = coercion_model.canonical_coercion(n, x)
return self._evalf_(n, x, parent(n))
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)
return None # leaves the expression unevaluated
def _eval_(self, a, z):
"""
EXAMPLES::
sage: struve_L(-2,0)
struve_L(-2, 0)
sage: struve_L(-1,0)
0
sage: struve_L(pi,0)
0
sage: struve_L(-1/2,x)
sqrt(2)*sqrt(1/(pi*x))*sinh(x)
sage: struve_L(1/2,1)
sqrt(2)*(cosh(1) - 1)/sqrt(pi)
sage: struve_L(2,x)
struve_L(2, x)
sage: struve_L(-3/2,x)
-bessel_I(3/2, x)
"""
from sage.symbolic.ring import SR
if z.is_zero() \
and (SR(a).is_numeric() or SR(a).is_constant()) \
and a.real() >= -1:
return ZZ(0)
if a == -Integer(1)/2:
from sage.functions.hyperbolic import sinh
return sqrt(2/(pi*z)) * sinh(z)
if a == Integer(1)/2:
from sage.functions.hyperbolic import cosh
return sqrt(2/(pi*z)) * (cosh(z)-1)
if a < 0 and not SR(a).is_integer() and SR(2*a).is_integer():
from sage.rings.rational_field import QQ
n = (a*(-2) - 1)/2
return Integer(-1)**n * bessel_I(n+QQ(1)/2, z)
def show(self, show_hyperboloid=True, **graphics_options):
r"""
Plot ``self``.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_geodesic import *
sage: g = HyperbolicPlane().HM().random_geodesic()
sage: g.show()
Graphics3d Object
"""
x = SR.var('x')
opts = self.graphics_options()
opts.update(graphics_options)
v1, u2 = [vector(k.coordinates()) for k in self.endpoints()]
# Lorentzian Gram Shmidt. The original vectors will be
# u1, u2 and the orthogonal ones will be v1, v2. Except
# v1 = u1, and I don't want to declare another variable,
# hence the odd naming convention above.
# We need the Lorentz dot product of v1 and u2.
v1_ldot_u2 = u2[0] * v1[0] + u2[1] * v1[1] - u2[2] * v1[2]
v2 = u2 + v1_ldot_u2 * v1
v2_norm = sqrt(v2[0]**2 + v2[1]**2 - v2[2]**2)
v2 = v2 / v2_norm
v2_ldot_u2 = u2[0] * v2[0] + u2[1] * v2[1] - u2[2] * v2[2]
# Now v1 and v2 are Lorentz orthogonal, and |v1| = -1, |v2|=1
# That is, v1 is unit timelike and v2 is unit spacelike.
# This means that cosh(x)*v1 + sinh(x)*v2 is unit timelike.
hyperbola = cosh(x) * v1 + sinh(x) * v2
endtime = arcsinh(v2_ldot_u2)
from sage.plot.plot3d.all import parametric_plot3d
pic = parametric_plot3d(hyperbola, (x, 0, endtime), **graphics_options)
if show_hyperboloid:
pic += self._model.get_background_graphic()
return pic
def show(self, show_hyperboloid=True, **graphics_options):
r"""
Plot ``self``.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_geodesic import *
sage: g = HyperbolicPlane().HM().random_geodesic()
sage: g.show()
Graphics3d Object
"""
x = SR.var('x')
opts = self.graphics_options()
opts.update(graphics_options)
v1, u2 = [vector(k.coordinates()) for k in self.endpoints()]
# Lorentzian Gram Shmidt. The original vectors will be
# u1, u2 and the orthogonal ones will be v1, v2. Except
# v1 = u1, and I don't want to declare another variable,
# hence the odd naming convention above.
# We need the Lorentz dot product of v1 and u2.
v1_ldot_u2 = u2[0]*v1[0] + u2[1]*v1[1] - u2[2]*v1[2]
v2 = u2 + v1_ldot_u2 * v1
v2_norm = sqrt(v2[0]**2 + v2[1]**2 - v2[2]**2)
v2 = v2 / v2_norm
v2_ldot_u2 = u2[0]*v2[0] + u2[1]*v2[1] - u2[2]*v2[2]
# Now v1 and v2 are Lorentz orthogonal, and |v1| = -1, |v2|=1
# That is, v1 is unit timelike and v2 is unit spacelike.
# This means that cosh(x)*v1 + sinh(x)*v2 is unit timelike.
hyperbola = cosh(x)*v1 + sinh(x)*v2
endtime = arcsinh(v2_ldot_u2)
from sage.plot.plot3d.all import parametric_plot3d
pic = parametric_plot3d(hyperbola, (x, 0, endtime), **graphics_options)
if show_hyperboloid:
pic += self._model.get_background_graphic()
return pic
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
if (not isinstance(n, Expression) and not isinstance(x, Expression) and
(is_inexact(n) or is_inexact(x))):
coercion_model = get_coercion_model()
n, x = coercion_model.canonical_coercion(n, x)
return self._evalf_(n, x, parent(n))
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)
return None # leaves the expression unevaluated
def _0f1(b, z):
F12 = cosh(2 * sqrt(z))
F32 = sinh(2 * sqrt(z)) / (2 * sqrt(z))
if 2 * b == 1:
return F12
if 2 * b == 3:
return F32
if 2 * b > 3:
return ((b - 2) * (b - 1) / z * (_0f1(b - 2, z) -
_0f1(b - 1, z)))
if 2 * b < 1:
return (_0f1(b + 1, z) + z / (b * (b + 1)) *
_0f1(b + 2, z))
raise ValueError
def _0f1(b, z):
F12 = cosh(2 * sqrt(z))
F32 = sinh(2 * sqrt(z)) / (2 * sqrt(z))
if 2 * b == 1:
return F12
if 2 * b == 3:
return F32
if 2 * b > 3:
return ((b - 2) * (b - 1) / z *
(_0f1(b - 2, z) - _0f1(b - 1, z)))
if 2 * b < 1:
return (_0f1(b + 1, z) + z / (b *
(b + 1)) * _0f1(b + 2, z))
raise ValueError
def _derivative_(self, z, diff_param=None):
r"""
The derivative of `\operatorname{Shi}(z)` is `\sinh(z)/z`.
EXAMPLES::
sage: x = var('x')
sage: f = sinh_integral(x)
sage: f.diff(x)
sinh(x)/x
sage: f = sinh_integral(ln(x))
sage: f.diff(x)
sinh(log(x))/(x*log(x))
"""
return sinh(z)/z
def _derivative_(self, z, diff_param=None):
r"""
The derivative of `\operatorname{Shi}(z)` is `\sinh(z)/z`.
EXAMPLES::
sage: x = var('x')
sage: f = sinh_integral(x)
sage: f.diff(x)
sinh(x)/x
sage: f = sinh_integral(ln(x))
sage: f.diff(x)
sinh(log(x))/(x*log(x))
"""
return sinh(z) / z
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)