def _evalf_(self, x, parent):
        """
        EXAMPLES::

            sage: erf(2).n()
            0.995322265018953
            sage: erf(2).n(200)
            0.99532226501895273416206925636725292861089179704006007673835
            sage: erf(pi - 1/2*I).n(100)
            1.0000111669099367825726058952 + 1.6332655417638522934072124548e-6*I

        TESTS:

        Check that PARI/GP through the GP interface gives the same answer::

            sage: gp.set_real_precision(59)  # random
            38
            sage: print gp.eval("1 - erfc(1)"); print erf(1).n(200);
            0.84270079294971486934122063508260925929606699796630290845994
            0.84270079294971486934122063508260925929606699796630290845994
        """
        try:
            prec = parent.prec()
        except AttributeError: # not a Sage parent
            prec = 0
        return parent(1) - parent(pari(x).erfc(prec))

    def _eval_(self, z):
        """
        EXAMPLES::

            sage: z = var('z')
            sage: sin_integral(z)
            sin_integral(z)
            sage: sin_integral(3.0)
            1.84865252799947
            sage: sin_integral(0)
            0

        """
        if not isinstance(z, Expression) and is_inexact(z):
            return self._evalf_(z, parent(z))

        # special case: z = 0
        if isinstance(z, Expression):
            if z.is_trivial_zero():
                return z
        else:
            if not z:
                return z

        return None # leaves the expression unevaluated

    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, x, y):
        """
        EXAMPLES::

            sage: gamma_inc(2.,0)
            1.00000000000000
            sage: gamma_inc(2,0)
            1
            sage: gamma_inc(1/2,2)
            -(erf(sqrt(2)) - 1)*sqrt(pi)
            sage: gamma_inc(1/2,1)
            -(erf(1) - 1)*sqrt(pi)
            sage: gamma_inc(1/2,0)
            sqrt(pi)
            sage: gamma_inc(x,0)
            gamma(x)
            sage: gamma_inc(1,2)
            e^(-2)
            sage: gamma_inc(0,2)
            -Ei(-2)
        """
        if not isinstance(x, Expression) and not isinstance(y, Expression) and \
               (is_inexact(x) or is_inexact(y)):
            x, y = coercion_model.canonical_coercion(x, y)
            return self._evalf_(x, y, parent(x))

        if y == 0:
            return gamma(x)
        if x == 1:
            return exp(-y)
        if x == 0:
            return -Ei(-y)
        if x == Rational(1)/2: #only for x>0
            return sqrt(pi)*(1-erf(sqrt(y)))
        return None

    def _evalf_(self, *args, **kwds):
        """
        Returns a numerical approximation of this function using
        Maxima.  Currently, this is limited to 53 bits of precision.

        EXAMPLES::

            sage: from sage.functions.special import MaximaFunction
            sage: f = MaximaFunction("jacobi_sn")
            sage: f(1/2,1/2)
            jacobi_sn(1/2, 1/2)
            sage: f(1/2,1/2).n()
            0.470750473655657

        TESTS::

            sage: f(1/2,1/2).n(150)
            Traceback (most recent call last):
            ...
            NotImplementedError: jacobi_sn not implemented for precision > 53
        """
        parent = kwds["parent"]
        if hasattr(parent, "prec") and parent.prec() > 53:
            raise NotImplementedError("%s not implemented for precision > 53" % self.name())
        _init()
        return parent(maxima("%s, numer" % self._maxima_init_evaled_(*args)))

    def _eval_(self, n, m, theta, phi, **kwargs):
        r"""
        TESTS::

            sage: x, y = var('x y')
            sage: spherical_harmonic(1, 2, x, y)
            0
            sage: spherical_harmonic(1, -2, x, y)
            0
            sage: spherical_harmonic(1/2, 2, x, y)
            spherical_harmonic(1/2, 2, x, y)
            sage: spherical_harmonic(3, 2, x, y)
            15/4*sqrt(7/30)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
            sage: spherical_harmonic(3, 2, 1, 2)
            15/4*sqrt(7/30)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
            sage: spherical_harmonic(3 + I, 2., 1, 2)
            -0.351154337307488 - 0.415562233975369*I
        """
        from sage.structure.coerce import parent

        cc = get_coercion_model().canonical_coercion
        coerced = cc(phi, cc(theta, cc(n, m)[0])[0])[0]
        if is_inexact(coerced) and not isinstance(coerced, Expression):
            return self._evalf_(n, m, theta, phi, parent=parent(coerced))
        elif n in ZZ and m in ZZ and n > -1:
            if abs(m) > n:
                return ZZ(0)
            return meval("spherical_harmonic({},{},{},{})".format(ZZ(n), ZZ(m), maxima(theta), maxima(phi)))
        return

 def _eval_(self, n, z ):
     """
     EXAMPLES::
     """
     # howto find a common parent for n and z here? 
     if not isinstance(z, Expression) and is_inexact(z):
         return self._evalf_(n, z, parent(z))
     return None

    def _evalf_(self, u, m, parent=None, algorithm=None):
        """
        EXAMPLES::

            sage: elliptic_eu(1,1).n()
            0.761594155955765
            sage: elliptic_eu(1,1).n(200)
            0.7615941559557648881194582...
        """
        R = parent or parent(z)
        return mpmath_utils.call(elliptic_eu_f, u, m, parent=R)

    def __call__(self, x, prec=None, coerce=True, hold=False):
        """
        Note that the ``prec`` argument is deprecated. The precision for
        the result is deduced from the precision of the input. Convert
        the input to a higher precision explicitly if a result with higher
        precision is desired.::

            sage: t = gamma(RealField(100)(2.5)); t
            1.3293403881791370204736256125
            sage: t.prec()
            100

            sage: gamma(6, prec=53)
            doctest:...: DeprecationWarning: The prec keyword argument is deprecated. Explicitly set the precision of the input, for example gamma(RealField(300)(1)), or use the prec argument to .n() for exact inputs, e.g., gamma(1).n(300), instead.
            120.000000000000

        TESTS::

            sage: gamma(pi,prec=100)
            2.2880377953400324179595889091

            sage: gamma(3/4,prec=100)
            1.2254167024651776451290983034
        """
        if prec is not None:
            from sage.misc.misc import deprecation
            deprecation("The prec keyword argument is deprecated. Explicitly set the precision of the input, for example gamma(RealField(300)(1)), or use the prec argument to .n() for exact inputs, e.g., gamma(1).n(300), instead.")
            import mpmath
            return mpmath_utils.call(mpmath.gamma, x, prec=prec)

        # this is a kludge to keep
        #     sage: Q.<i> = NumberField(x^2+1)
        #     sage: gamma(i)
        # working, since number field elements cannot be coerced into SR
        # without specifying an explicit embedding into CC any more
        try:
            res = GinacFunction.__call__(self, x, coerce=coerce, hold=hold)
        except TypeError, err:
            # the __call__() method returns a TypeError for fast float arguments
            # as well, we only proceed if the error message says that
            # the arguments cannot be coerced to SR
            if not str(err).startswith("cannot coerce"):
                raise

            from sage.misc.misc import deprecation
            deprecation("Calling symbolic functions with arguments that cannot be coerced into symbolic expressions is deprecated.")
            parent = RR if prec is None else RealField(prec)
            try:
                x = parent(x)
            except (ValueError, TypeError):
                x = parent.complex_field()(x)
            res = GinacFunction.__call__(self, x, coerce=coerce, hold=hold)

    def eval_formula(self, n, x):
        """
        Evaluate ``chebyshev_T`` using an explicit formula.
        See [ASHandbook]_ 227 (p. 782) for details for the recurions.
        See also [EffCheby]_ for fast evaluation techniques.

        INPUT:

        - ``n`` -- an integer

        - ``x`` -- a value to evaluate the polynomial at (this can be
          any ring element)

        EXAMPLES::

            sage: chebyshev_T.eval_formula(-1,x)
            x
            sage: chebyshev_T.eval_formula(0,x)
            1
            sage: chebyshev_T.eval_formula(1,x)
            x
            sage: chebyshev_T.eval_formula(2,0.1) == chebyshev_T._evalf_(2,0.1)
            True
            sage: chebyshev_T.eval_formula(10,x)
            512*x^10 - 1280*x^8 + 1120*x^6 - 400*x^4 + 50*x^2 - 1
            sage: chebyshev_T.eval_algebraic(10,x).expand()
            512*x^10 - 1280*x^8 + 1120*x^6 - 400*x^4 + 50*x^2 - 1
        """
        if n < 0:
            return self.eval_formula(-n, x)
        elif n == 0:
            return parent(x).one()

        res = parent(x).zero()
        for j in xrange(0, n // 2 + 1):
            f = factorial(n - 1 - j) / factorial(j) / factorial(n - 2 * j)
            res += (-1) ** j * (2 * x) ** (n - 2 * j) * f
        res *= n / 2
        return res

    def eval_algebraic(self, n, x):
        """
        Evaluate :class:`chebyshev_U` as polynomial, using a recursive
        formula.

        INPUT:

        - ``n`` -- an integer

        - ``x`` -- a value to evaluate the polynomial at (this can be
          any ring element)

        EXAMPLES::

            sage: chebyshev_U.eval_algebraic(5,x)
            -2*((2*x + 1)*(2*x - 1)*x - 4*(2*x^2 - 1)*x)*(2*x + 1)*(2*x - 1)
            sage: parent(chebyshev_U(3, Mod(8,9)))
            Ring of integers modulo 9
            sage: parent(chebyshev_U(3, Mod(1,9)))
            Ring of integers modulo 9
            sage: chebyshev_U(-3,x) + chebyshev_U(1,x)
            0
            sage: chebyshev_U(-1,Mod(5,8))
            0
            sage: parent(chebyshev_U(-1,Mod(5,8)))
            Ring of integers modulo 8
            sage: R.<t> = ZZ[]
            sage: chebyshev_U.eval_algebraic(-2, t)
            -1
            sage: chebyshev_U.eval_algebraic(-1, t)
            0
            sage: chebyshev_U.eval_algebraic(0, t)
            1
            sage: chebyshev_U.eval_algebraic(1, t)
            2*t
            sage: n = 97; x = RIF(pi/n)
            sage: chebyshev_U(n-1, cos(x)).contains_zero()
            True
            sage: R.<t> = Zp(2, 6, 'capped-abs')[]
            sage: chebyshev_U(10^6+1, t)
            (2 + O(2^6))*t + (O(2^6))
        """
        if n == -1:
            return parent(x).zero()
        if n < 0:
            return -self._eval_recursive_(-n - 2, x)[0]
        return self._eval_recursive_(n, x)[0]

 def _eval_(self, a, b, z, **kwargs):
     """
     EXAMPLES::
     
         sage: hypergeometric([], [], 0)
         1
     """
     if not isinstance(a,tuple) or not isinstance(b,tuple):
         raise ValueError('First two parameters must be of type list.')
     coercion_model = get_coercion_model()
     co = reduce(lambda x, y: coercion_model.canonical_coercion(x, y)[0],
                 a + b + (z,))
     if is_inexact(co) and not isinstance(co, Expression):
         from sage.structure.coerce import parent
         return self._evalf_(a, b, z, parent=parent(co))
     if not isinstance(z, Expression) and z == 0:  # Expression is excluded
         return Integer(1)                         # to avoid call to Maxima
     return

    def _eval_recursive_(self, n, x, both=False):
        """
        If ``both=True``, compute ``(U(n,x), U(n-1,x))`` using a
        recursive formula.
        If ``both=False``, return instead a tuple ``(U(n,x), False)``.

        EXAMPLES::

            sage: chebyshev_U._eval_recursive_(3, x)
            (4*((2*x + 1)*(2*x - 1) - 2*x^2)*x, False)
            sage: chebyshev_U._eval_recursive_(3, x, True)
            (4*((2*x + 1)*(2*x - 1) - 2*x^2)*x, ((2*x + 1)*(2*x - 1) + 2*x)*((2*x + 1)*(2*x - 1) - 2*x))
        """
        if n == 0:
            return parent(x).one(), 2 * x

        assert n >= 1
        a, b = self._eval_recursive_((n - 1) // 2, x, True)
        if n % 2 == 0:
            return (b + a) * (b - a), both and 2 * b * (x * b - a)
        else:
            return 2 * a * (b - x * a), both and (b + a) * (b - a)