Пример #1
0
 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])
Пример #2
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    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
Пример #3
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    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)
Пример #4
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    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
Пример #5
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    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
Пример #6
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    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
Пример #7
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 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
Пример #8
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 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
Пример #9
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    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
Пример #10
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    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
Пример #11
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    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)
Пример #12
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    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)