""" Integral Transforms """

from functools import reduce, wraps

from diofant.core import S, sympify
from diofant.core.function import Function
from diofant.core.numbers import oo, Integer, Rational
from diofant.core.symbol import Dummy
from diofant.integrals import integrate, Integral
from diofant.integrals.meijerint import _dummy
from diofant.logic.boolalg import to_cnf, conjuncts, disjuncts, Or, And
from diofant.simplify import simplify
from diofant.utilities import default_sort_key
from diofant.matrices.matrices import MatrixBase


##########################################################################
# Helpers / Utilities
##########################################################################


class IntegralTransformError(NotImplementedError):
    """
    Exception raised in relation to problems computing transforms.

    This class is mostly used internally; if integrals cannot be computed
    objects representing unevaluated transforms are usually returned.

    The hint ``needeval=True`` can be used to disable returning transform
    objects, and instead raise this exception if an integral cannot be
    computed.
    """
    def __init__(self, transform, function, msg):
        super(IntegralTransformError, self).__init__(
            "%s Transform could not be computed: %s." % (transform, msg))
        self.function = function


class IntegralTransform(Function):
    """
    Base class for integral transforms.

    This class represents unevaluated transforms.

    To implement a concrete transform, derive from this class and implement
    the ``_compute_transform(f, x, s, **hints)`` and ``_as_integral(f, x, s)``
    functions. If the transform cannot be computed, raise IntegralTransformError.

    Also set ``cls._name``.

    Implement self._collapse_extra if your function returns more than just a
    number and possibly a convergence condition.
    """

    @property
    def function(self):
        """ The function to be transformed. """
        return self.args[0]

    @property
    def function_variable(self):
        """ The dependent variable of the function to be transformed. """
        return self.args[1]

    @property
    def transform_variable(self):
        """ The independent transform variable. """
        return self.args[2]

    @property
    def free_symbols(self):
        """
        This method returns the symbols that will exist when the transform
        is evaluated.
        """
        return self.function.free_symbols.union({self.transform_variable}) \
            - {self.function_variable}

    def _compute_transform(self, f, x, s, **hints):
        raise NotImplementedError

    def _as_integral(self, f, x, s):
        raise NotImplementedError

    def _collapse_extra(self, extra):
        cond = And(*extra)
        if cond == S.false:
            raise IntegralTransformError(self.__class__.name, None, '')

    def doit(self, **hints):
        """
        Try to evaluate the transform in closed form.

        This general function handles linearity, but apart from that leaves
        pretty much everything to _compute_transform.

        Standard hints are the following:

        - ``simplify``: whether or not to simplify the result
        - ``noconds``: if True, don't return convergence conditions
        - ``needeval``: if True, raise IntegralTransformError instead of
                        returning IntegralTransform objects

        The default values of these hints depend on the concrete transform,
        usually the default is
        ``(simplify, noconds, needeval) = (True, False, False)``.
        """
        from diofant import Add, expand_mul, Mul
        from diofant.core.function import AppliedUndef
        needeval = hints.pop('needeval', False)
        try_directly = not any(func.has(self.function_variable)
                               for func in self.function.atoms(AppliedUndef))
        if try_directly:
            try:
                return self._compute_transform(self.function,
                    self.function_variable, self.transform_variable, **hints)
            except IntegralTransformError:
                pass

        fn = self.function
        if not fn.is_Add:
            fn = expand_mul(fn)

        if fn.is_Add:
            hints['needeval'] = needeval
            res = [self.__class__(*([x] + list(self.args[1:]))).doit(**hints)
                   for x in fn.args]
            extra = []
            ress = []
            for x in res:
                if not isinstance(x, tuple):
                    x = [x]
                ress.append(x[0])
                if len(x) > 1:
                    extra += [x[1:]]
            res = Add(*ress)
            if not extra:
                return res
            try:
                extra = self._collapse_extra(extra)
                return tuple([res]) + tuple(extra)
            except IntegralTransformError:
                pass

        if needeval:
            raise IntegralTransformError(
                self.__class__._name, self.function, 'needeval')

        # TODO handle derivatives etc

        # pull out constant coefficients
        coeff, rest = fn.as_coeff_mul(self.function_variable)
        return coeff*self.__class__(*([Mul(*rest)] + list(self.args[1:])))

    @property
    def as_integral(self):
        return self._as_integral(self.function, self.function_variable,
                                 self.transform_variable)

    def _eval_rewrite_as_Integral(self, *args):
        return self.as_integral

from diofant.solvers.inequalities import solve_univariate_inequality


def _simplify(expr, doit):
    from diofant import powdenest, piecewise_fold
    if doit:
        return simplify(powdenest(piecewise_fold(expr), polar=True))
    return expr


def _noconds_(default):
    """
    This is a decorator generator for dropping convergence conditions.

    Suppose you define a function ``transform(*args)`` which returns a tuple of
    the form ``(result, cond1, cond2, ...)``.

    Decorating it ``@_noconds_(default)`` will add a new keyword argument
    ``noconds`` to it. If ``noconds=True``, the return value will be altered to
    be only ``result``, whereas if ``noconds=False`` the return value will not
    be altered.

    The default value of the ``noconds`` keyword will be ``default`` (i.e. the
    argument of this function).
    """
    def make_wrapper(func):
        @wraps(func)
        def wrapper(*args, **kwargs):
            noconds = kwargs.pop('noconds', default)
            res = func(*args, **kwargs)
            if noconds:
                return res[0]
            return res
        return wrapper
    return make_wrapper
_noconds = _noconds_(False)


##########################################################################
# Mellin Transform
##########################################################################

def _default_integrator(f, x):
    return integrate(f, (x, 0, oo))


@_noconds
def _mellin_transform(f, x, s_, integrator=_default_integrator, simplify=True):
    """ Backend function to compute Mellin transforms. """
    from diofant import re, Max, Min, count_ops
    # We use a fresh dummy, because assumptions on s might drop conditions on
    # convergence of the integral.
    s = _dummy('s', 'mellin-transform', f)
    F = integrator(x**(s - 1) * f, x)

    if not F.has(Integral):
        return _simplify(F.subs(s, s_), simplify), (-oo, oo), True

    if not F.is_Piecewise:
        raise IntegralTransformError('Mellin', f, 'could not compute integral')

    F, cond = F.args[0]
    if F.has(Integral):
        raise IntegralTransformError(
            'Mellin', f, 'integral in unexpected form')

    def process_conds(cond):
        """
        Turn ``cond`` into a strip (a, b), and auxiliary conditions.
        """
        a = -oo
        b = oo
        aux = True
        conds = conjuncts(to_cnf(cond))
        t = Dummy('t', extended_real=True)
        for c in conds:
            a_ = oo
            b_ = -oo
            aux_ = []
            for d in disjuncts(c):
                d_ = d.replace(
                    re, lambda x: x.as_real_imag()[0]).subs(re(s), t)
                if not d.is_Relational or \
                    d.rel_op not in ('>', '>=', '<', '<=') \
                        or d_.has(s) or not d_.has(t):
                    aux_ += [d]
                    continue
                soln = solve_univariate_inequality(d_, t)
                t_ = Dummy("t", real=True)
                soln = soln.subs(t, t_).subs(t_, t)
                if not soln.is_Relational or \
                        soln.rel_op not in ('>', '>=', '<', '<='):
                    aux_ += [d]
                    continue
                if soln.lts == t:
                    b_ = Max(soln.gts, b_)
                else:
                    a_ = Min(soln.lts, a_)
            if a_ != oo and a_ != b:
                a = Max(a_, a)
            elif b_ != -oo and b_ != a:
                b = Min(b_, b)
            else:
                aux = And(aux, Or(*aux_))
        return a, b, aux

    conds = [process_conds(c) for c in disjuncts(cond)]
    conds = [x for x in conds if x[2] is not S.false]
    conds.sort(key=lambda x: (x[0] - x[1], count_ops(x[2])))

    if not conds:
        raise IntegralTransformError('Mellin', f, 'no convergence found')

    a, b, aux = conds[0]
    return _simplify(F.subs(s, s_), simplify), (a, b), aux


class MellinTransform(IntegralTransform):
    """
    Class representing unevaluated Mellin transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute Mellin transforms, see the :func:`mellin_transform`
    docstring.
    """

    _name = 'Mellin'

    def _compute_transform(self, f, x, s, **hints):
        return _mellin_transform(f, x, s, **hints)

    def _as_integral(self, f, x, s):
        return Integral(f*x**(s - 1), (x, 0, oo))

    def _collapse_extra(self, extra):
        from diofant import Max, Min
        a = []
        b = []
        cond = []
        for (sa, sb), c in extra:
            a += [sa]
            b += [sb]
            cond += [c]
        res = (Max(*a), Min(*b)), And(*cond)
        if (res[0][0] >= res[0][1]) is S.true or res[1] is S.false:
            raise IntegralTransformError(
                'Mellin', None, 'no combined convergence.')
        return res


def mellin_transform(f, x, s, **hints):
    r"""
    Compute the Mellin transform `F(s)` of `f(x)`,

    .. math :: F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x.

    For all "sensible" functions, this converges absolutely in a strip
      `a < \operatorname{Re}(s) < b`.

    The Mellin transform is related via change of variables to the Fourier
    transform, and also to the (bilateral) Laplace transform.

    This function returns ``(F, (a, b), cond)``
    where ``F`` is the Mellin transform of ``f``, ``(a, b)`` is the fundamental strip
    (as above), and ``cond`` are auxiliary convergence conditions.

    If the integral cannot be computed in closed form, this function returns
    an unevaluated :class:`~diofant.integrals.transforms.MellinTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`. If ``noconds=False``,
    then only `F` will be returned (i.e. not ``cond``, and also not the strip
    ``(a, b)``).

    >>> from diofant.integrals.transforms import mellin_transform
    >>> from diofant import exp
    >>> from diofant.abc import x, s
    >>> mellin_transform(exp(-x), x, s)
    (gamma(s), (0, oo), True)

    See Also
    ========

    inverse_mellin_transform, laplace_transform, fourier_transform
    hankel_transform, inverse_hankel_transform
    """
    return MellinTransform(f, x, s).doit(**hints)


def _rewrite_sin(m_n, s, a, b):
    """
    Re-write the sine function ``sin(m*s + n)`` as gamma functions, compatible
    with the strip (a, b).

    Return ``(gamma1, gamma2, fac)`` so that ``f == fac/(gamma1 * gamma2)``.

    >>> from diofant.integrals.transforms import _rewrite_sin
    >>> from diofant import pi, Rational
    >>> from diofant.abc import s
    >>> _rewrite_sin((pi, 0), s, 0, 1)
    (gamma(s), gamma(-s + 1), pi)
    >>> _rewrite_sin((pi, 0), s, 1, 0)
    (gamma(s - 1), gamma(-s + 2), -pi)
    >>> _rewrite_sin((pi, 0), s, -1, 0)
    (gamma(s + 1), gamma(-s), -pi)
    >>> _rewrite_sin((pi, pi/2), s, Rational(1, 2), Rational(3, 2))
    (gamma(s - 1/2), gamma(-s + 3/2), -pi)
    >>> _rewrite_sin((pi, pi), s, 0, 1)
    (gamma(s), gamma(-s + 1), -pi)
    >>> _rewrite_sin((2*pi, 0), s, 0, Rational(1, 2))
    (gamma(2*s), gamma(-2*s + 1), pi)
    >>> _rewrite_sin((2*pi, 0), s, Rational(1, 2), 1)
    (gamma(2*s - 1), gamma(-2*s + 2), -pi)
    """
    # (This is a separate function because it is moderately complicated,
    #  and I want to doctest it.)
    # We want to use pi/sin(pi*x) = gamma(x)*gamma(1-x).
    # But there is one comlication: the gamma functions determine the
    # inegration contour in the definition of the G-function. Usually
    # it would not matter if this is slightly shifted, unless this way
    # we create an undefined function!
    # So we try to write this in such a way that the gammas are
    # eminently on the right side of the strip.
    from diofant import expand_mul, pi, ceiling, gamma
    m, n = m_n

    m = expand_mul(m/pi)
    n = expand_mul(n/pi)
    r = ceiling(-m*a - n.as_real_imag()[0])  # Don't use re(n), does not expand
    return gamma(m*s + n + r), gamma(1 - n - r - m*s), (-1)**r*pi


class MellinTransformStripError(ValueError):
    """
    Exception raised by _rewrite_gamma. Mainly for internal use.
    """
    pass


def _rewrite_gamma(f, s, a, b):
    """
    Try to rewrite the product f(s) as a product of gamma functions,
    so that the inverse Mellin transform of f can be expressed as a meijer
    G function.

    Return (an, ap), (bm, bq), arg, exp, fac such that
    G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse Mellin transform of f(s).

    Raises IntegralTransformError or MellinTransformStripError on failure.

    It is asserted that f has no poles in the fundamental strip designated by
    (a, b). One of a and b is allowed to be None. The fundamental strip is
    important, because it determines the inversion contour.

    This function can handle exponentials, linear factors, trigonometric
    functions.

    This is a helper function for inverse_mellin_transform that will not
    attempt any transformations on f.

    >>> from diofant.integrals.transforms import _rewrite_gamma
    >>> from diofant.abc import s
    >>> from diofant import oo
    >>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo)
    (([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1)
    >>> _rewrite_gamma((s-1)**2, s, -oo, oo)
    (([], [1, 1]), ([2, 2], []), 1, 1, 1)

    Importance of the fundamental strip:

    >>> _rewrite_gamma(1/s, s, 0, oo)
    (([1], []), ([], [0]), 1, 1, 1)
    >>> _rewrite_gamma(1/s, s, None, oo)
    (([1], []), ([], [0]), 1, 1, 1)
    >>> _rewrite_gamma(1/s, s, 0, None)
    (([1], []), ([], [0]), 1, 1, 1)
    >>> _rewrite_gamma(1/s, s, -oo, 0)
    (([], [1]), ([0], []), 1, 1, -1)
    >>> _rewrite_gamma(1/s, s, None, 0)
    (([1], []), ([], [0]), 1, 1, 1)
    >>> _rewrite_gamma(1/s, s, -oo, None)
    (([], [1]), ([0], []), 1, 1, -1)

    >>> _rewrite_gamma(2**(-s+3), s, -oo, oo)
    (([], []), ([], []), 1/2, 1, 8)
    """
    from itertools import repeat
    from diofant import (Poly, gamma, Mul, re, RootOf, expand,
                       roots, ilcm, pi, sin, cos, tan, cot, igcd, exp_polar)
    # Our strategy will be as follows:
    # 1) Guess a constant c such that the inversion integral should be
    #    performed wrt s'=c*s (instead of plain s). Write s for s'.
    # 2) Process all factors, rewrite them independently as gamma functions in
    #    argument s, or exponentials of s.
    # 3) Try to transform all gamma functions s.t. they have argument
    #    a+s or a-s.
    # 4) Check that the resulting G function parameters are valid.
    # 5) Combine all the exponentials.

    a_, b_ = sympify([a, b])

    def left(c, is_numer):
        """
        Decide whether pole at c lies to the left of the fundamental strip.
        """
        # heuristically, this is the best chance for us to solve the inequalities
        c = expand(re(c))
        if a_ is None:
            return b_ >= c
        if b_ is None:
            return a_ >= c
        if (c >= b_) is S.true:
            return S.false
        if (c <= a_) is S.true:
            return S.true
        if is_numer:
            return
        if a_.free_symbols or b_.free_symbols or c.free_symbols:
            return  # XXX
            # raise IntegralTransformError('Inverse Mellin', f,
            #                     'Could not determine position of singularity %s'
            #                     ' relative to fundamental strip' % c)
        raise MellinTransformStripError('Pole inside critical strip?')

    # 1)
    s_multipliers = []
    for g in f.atoms(gamma):
        if not g.has(s):
            continue
        arg = g.args[0]
        if arg.is_Add:
            arg = arg.as_independent(s)[1]
        coeff, _ = arg.as_coeff_mul(s)
        s_multipliers += [coeff]
    for g in f.atoms(sin, cos, tan, cot):
        if not g.has(s):
            continue
        arg = g.args[0]
        if arg.is_Add:
            arg = arg.as_independent(s)[1]
        coeff, _ = arg.as_coeff_mul(s)
        s_multipliers += [coeff/pi]
    s_multipliers = [abs(x) if x.is_extended_real else x for x in s_multipliers]
    common_coefficient = Integer(1)
    for x in s_multipliers:
        if not x.is_Rational:
            common_coefficient = x
            break
    s_multipliers = [x/common_coefficient for x in s_multipliers]
    if (any(not x.is_Rational for x in s_multipliers) or
            not common_coefficient.is_extended_real):
        raise IntegralTransformError("Gamma", None, "Nonrational multiplier")
    s_multiplier = common_coefficient/reduce(ilcm, [Integer(x.q)
                                             for x in s_multipliers], Integer(1))
    if s_multiplier == common_coefficient:
        if len(s_multipliers) == 0:
            s_multiplier = common_coefficient
        else:
            s_multiplier = common_coefficient \
                * reduce(igcd, [Integer(x.p) for x in s_multipliers])

    exponent = Integer(1)
    fac = Integer(1)
    f = f.subs(s, s/s_multiplier)
    fac /= s_multiplier
    exponent = 1/s_multiplier
    if a_ is not None:
        a_ *= s_multiplier
    if b_ is not None:
        b_ *= s_multiplier

    # 2)
    numer, denom = f.as_numer_denom()
    numer = Mul.make_args(numer)
    denom = Mul.make_args(denom)
    args = list(zip(numer, repeat(True))) + list(zip(denom, repeat(False)))

    facs = []
    dfacs = []
    # *_gammas will contain pairs (a, c) representing Gamma(a*s + c)
    numer_gammas = []
    denom_gammas = []
    # exponentials will contain bases for exponentials of s
    exponentials = []

    def exception(fact):
        return IntegralTransformError("Inverse Mellin", f, "Unrecognised form '%s'." % fact)
    while args:
        fact, is_numer = args.pop()
        if is_numer:
            ugammas, lgammas = numer_gammas, denom_gammas
            ufacs, lfacs = facs, dfacs
        else:
            ugammas, lgammas = denom_gammas, numer_gammas
            ufacs, lfacs = dfacs, facs

        def linear_arg(arg):
            """ Test if arg is of form a*s+b, raise exception if not. """
            if not arg.is_polynomial(s):
                raise exception(fact)
            p = Poly(arg, s)
            if p.degree() != 1:
                raise exception(fact)
            return p.all_coeffs()

        # constants
        if not fact.has(s):
            ufacs += [fact]
        # exponentials
        elif fact.is_Pow:
            if fact.is_Pow and fact.base is not S.Exp1:
                base = fact.base
                exp = fact.exp
            else:
                base = exp_polar(1)
                exp = fact.exp
            if exp.is_Integer:
                cond = is_numer
                if exp < 0:
                    cond = not cond
                args += [(base, cond)]*abs(exp)
                continue
            elif not base.has(s):
                a, b = linear_arg(exp)
                if not is_numer:
                    base = 1/base
                exponentials += [base**a]
                facs += [base**b]
            else:
                raise exception(fact)
        # linear factors
        elif fact.is_polynomial(s):
            p = Poly(fact, s)
            if p.degree() != 1:
                # We completely factor the poly. For this we need the roots.
                # Now roots() only works in some cases (low degree), and RootOf
                # only works without parameters. So try both...
                coeff = p.LT()[1]
                rs = roots(p, s)
                if len(rs) != p.degree():
                    rs = RootOf.all_roots(p)
                ufacs += [coeff]
                args += [(s - c, is_numer) for c in rs]
                continue
            a, c = p.all_coeffs()
            ufacs += [a]
            c /= -a
            # Now need to convert s - c
            if left(c, is_numer):
                ugammas += [(Integer(1), -c + 1)]
                lgammas += [(Integer(1), -c)]
            else:
                ufacs += [-1]
                ugammas += [(Integer(-1), c + 1)]
                lgammas += [(Integer(-1), c)]
        elif isinstance(fact, gamma):
            a, b = linear_arg(fact.args[0])
            if is_numer:
                if (a > 0 and (left(-b/a, is_numer) is S.false)) or \
                   (a < 0 and (left(-b/a, is_numer) is S.true)):
                    raise NotImplementedError(
                        'Gammas partially over the strip.')
            ugammas += [(a, b)]
        elif isinstance(fact, sin):
            # We try to re-write all trigs as gammas. This is not in
            # general the best strategy, since sometimes this is impossible,
            # but rewriting as exponentials would work. However trig functions
            # in inverse mellin transforms usually all come from simplifying
            # gamma terms, so this should work.
            a = fact.args[0]
            if is_numer:
                # No problem with the poles.
                gamma1, gamma2, fac_ = gamma(a/pi), gamma(1 - a/pi), pi
            else:
                gamma1, gamma2, fac_ = _rewrite_sin(linear_arg(a), s, a_, b_)
            args += [(gamma1, not is_numer), (gamma2, not is_numer)]
            ufacs += [fac_]
        elif isinstance(fact, tan):
            a = fact.args[0]
            args += [(sin(a, evaluate=False), is_numer),
                     (sin(pi/2 - a, evaluate=False), not is_numer)]
        elif isinstance(fact, cos):
            a = fact.args[0]
            args += [(sin(pi/2 - a, evaluate=False), is_numer)]
        elif isinstance(fact, cot):
            a = fact.args[0]
            args += [(sin(pi/2 - a, evaluate=False), is_numer),
                     (sin(a, evaluate=False), not is_numer)]
        else:
            raise exception(fact)

    fac *= Mul(*facs)/Mul(*dfacs)

    # 3)
    an, ap, bm, bq = [], [], [], []
    for gammas, plus, minus, is_numer in [(numer_gammas, an, bm, True),
                                          (denom_gammas, bq, ap, False)]:
        while gammas:
            a, c = gammas.pop()
            if a != -1 and a != +1:
                # We use the gamma function multiplication theorem.
                p = abs(sympify(a))
                newa = a/p
                newc = c/p
                if not a.is_Integer:
                    raise TypeError("a is not an integer")
                for k in range(p):
                    gammas += [(newa, newc + k/p)]
                if is_numer:
                    fac *= (2*pi)**((1 - p)/2) * p**(c - Rational(1, 2))
                    exponentials += [p**a]
                else:
                    fac /= (2*pi)**((1 - p)/2) * p**(c - Rational(1, 2))
                    exponentials += [p**(-a)]
                continue
            if a == +1:
                plus.append(1 - c)
            else:
                minus.append(c)

    # 4)
    # TODO

    # 5)
    arg = Mul(*exponentials)

    # for testability, sort the arguments
    an.sort(key=default_sort_key)
    ap.sort(key=default_sort_key)
    bm.sort(key=default_sort_key)
    bq.sort(key=default_sort_key)

    return (an, ap), (bm, bq), arg, exponent, fac


@_noconds_(True)
def _inverse_mellin_transform(F, s, x_, strip, as_meijerg=False):
    """ A helper for the real inverse_mellin_transform function, this one here
        assumes x to be real and positive. """
    from diofant import (expand, expand_mul, hyperexpand, meijerg,
                       arg, pi, re, factor, Heaviside, gamma, Add)
    x = _dummy('t', 'inverse-mellin-transform', F, positive=True)
    # Actually, we won't try integration at all. Instead we use the definition
    # of the Meijer G function as a fairly general inverse mellin transform.
    F = F.rewrite(gamma)
    for g in [factor(F), expand_mul(F), expand(F)]:
        if g.is_Add:
            # do all terms separately
            ress = [_inverse_mellin_transform(G, s, x, strip, as_meijerg,
                                              noconds=False)
                    for G in g.args]
            conds = [p[1] for p in ress]
            ress = [p[0] for p in ress]
            res = Add(*ress)
            if not as_meijerg:
                res = factor(res, gens=res.atoms(Heaviside))
            return res.subs(x, x_), And(*conds)

        try:
            a, b, C, e, fac = _rewrite_gamma(g, s, strip[0], strip[1])
        except IntegralTransformError:
            continue
        G = meijerg(a, b, C/x**e)
        if as_meijerg:
            h = G
        else:
            try:
                h = hyperexpand(G)
            except NotImplementedError as detail:
                raise IntegralTransformError(
                    'Inverse Mellin', F, 'Could not calculate integral')

            if h.is_Piecewise and len(h.args) == 3:
                # XXX we break modularity here!
                h = Heaviside(x - abs(C))*h.args[0].args[0] \
                    + Heaviside(abs(C) - x)*h.args[1].args[0]
        # We must ensure that the intgral along the line we want converges,
        # and return that value.
        # See [L], 5.2
        cond = [abs(arg(G.argument)) < G.delta*pi]
        # Note: we allow ">=" here, this corresponds to convergence if we let
        # limits go to oo symetrically. ">" corresponds to absolute convergence.
        cond += [And(Or(len(G.ap) != len(G.bq), 0 >= re(G.nu) + 1),
                     abs(arg(G.argument)) == G.delta*pi)]
        cond = Or(*cond)
        if cond == S.false:
            raise IntegralTransformError(
                'Inverse Mellin', F, 'does not converge')
        return (h*fac).subs(x, x_), cond

    raise IntegralTransformError('Inverse Mellin', F, '')

_allowed = None


class InverseMellinTransform(IntegralTransform):
    """
    Class representing unevaluated inverse Mellin transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute inverse Mellin transforms, see the
    :func:`inverse_mellin_transform` docstring.
    """

    _name = 'Inverse Mellin'
    _none_sentinel = Dummy('None')
    _c = Dummy('c')

    def __new__(cls, F, s, x, a, b, **opts):
        if a is None:
            a = InverseMellinTransform._none_sentinel
        if b is None:
            b = InverseMellinTransform._none_sentinel
        return IntegralTransform.__new__(cls, F, s, x, a, b, **opts)

    @property
    def fundamental_strip(self):
        a, b = self.args[3], self.args[4]
        if a is InverseMellinTransform._none_sentinel:
            a = None
        if b is InverseMellinTransform._none_sentinel:
            b = None
        return a, b

    def _compute_transform(self, F, s, x, **hints):
        from diofant import postorder_traversal
        global _allowed
        if _allowed is None:
            from diofant import (
                exp, gamma, sin, cos, tan, cot, cosh, sinh, tanh,
                coth, factorial, rf)
            _allowed = {exp, gamma, sin, cos, tan, cot, cosh, sinh, tanh, coth,
                        factorial, rf}
        for f in postorder_traversal(F):
            if f.is_Function and f.has(s) and f.func not in _allowed:
                raise IntegralTransformError('Inverse Mellin', F,
                                     'Component %s not recognised.' % f)
        strip = self.fundamental_strip
        return _inverse_mellin_transform(F, s, x, strip, **hints)

    def _as_integral(self, F, s, x):
        from diofant import I
        c = self.__class__._c
        return Integral(F*x**(-s), (s, c - I*oo, c + I*oo))


def inverse_mellin_transform(F, s, x, strip, **hints):
    r"""
    Compute the inverse Mellin transform of `F(s)` over the fundamental
    strip given by ``strip=(a, b)``.

    This can be defined as

    .. math:: f(x) = \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s,

    for any `c` in the fundamental strip. Under certain regularity
    conditions on `F` and/or `f`,
    this recovers `f` from its Mellin transform `F`
    (and vice versa), for positive real `x`.

    One of `a` or `b` may be passed as ``None``; a suitable `c` will be
    inferred.

    If the integral cannot be computed in closed form, this function returns
    an unevaluated :class:`~diofant.integrals.transforms.InverseMellinTransform` object.

    Note that this function will assume x to be positive and real, regardless
    of the diofant assumptions!

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.

    >>> from diofant.integrals.transforms import inverse_mellin_transform
    >>> from diofant import oo, gamma
    >>> from diofant.abc import x, s
    >>> inverse_mellin_transform(gamma(s), s, x, (0, oo))
    E**(-x)

    The fundamental strip matters:

    >>> f = 1/(s**2 - 1)
    >>> inverse_mellin_transform(f, s, x, (-oo, -1))
    (x/2 - 1/(2*x))*Heaviside(x - 1)
    >>> inverse_mellin_transform(f, s, x, (-1, 1))
    -x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
    >>> inverse_mellin_transform(f, s, x, (1, oo))
    (-x/2 + 1/(2*x))*Heaviside(-x + 1)

    See Also
    ========

    mellin_transform
    hankel_transform, inverse_hankel_transform
    """
    return InverseMellinTransform(F, s, x, strip[0], strip[1]).doit(**hints)


##########################################################################
# Laplace Transform
##########################################################################

def _simplifyconds(expr, s, a):
    r"""
    Naively simplify some conditions occuring in ``expr``, given that `\operatorname{Re}(s) > a`.

    >>> from diofant.integrals.transforms import _simplifyconds as simp
    >>> from diofant.abc import x
    >>> from diofant import sympify as S
    >>> simp(abs(x**2) < 1, x, 1)
    False
    >>> simp(abs(x**2) < 1, x, 2)
    False
    >>> simp(abs(x**2) < 1, x, 0)
    Abs(x**2) < 1
    >>> simp(abs(1/x**2) < 1, x, 1)
    True
    >>> simp(Integer(1) < abs(x), x, 1)
    True
    >>> simp(Integer(1) < abs(1/x), x, 1)
    False

    >>> from diofant import Ne
    >>> simp(Ne(1, x**3), x, 1)
    True
    >>> simp(Ne(1, x**3), x, 2)
    True
    >>> simp(Ne(1, x**3), x, 0)
    Ne(1, x**3)
    """
    from diofant.core.relational import ( StrictGreaterThan, StrictLessThan,
        Unequality )
    from diofant import Abs

    def power(ex):
        if ex == s:
            return 1
        if ex.is_Pow and ex.base == s:
            return ex.exp
        return

    def bigger(ex1, ex2):
        """ Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|.
            Else return None. """
        if ex1.has(s) and ex2.has(s):
            return
        if ex1.func is Abs:
            ex1 = ex1.args[0]
        if ex2.func is Abs:
            ex2 = ex2.args[0]
        if ex1.has(s):
            return bigger(1/ex2, 1/ex1)
        n = power(ex2)
        if n is None:
            return
        try:
            if n > 0 and (abs(ex1) <= abs(a)**n) is S.true:
                return False
            if n < 0 and (abs(ex1) >= abs(a)**n) is S.true:
                return True
        except TypeError:
            pass

    def replie(x, y):
        """ simplify x < y """
        if not (x.is_positive or x.func is Abs) \
                or not (y.is_positive or y.func is Abs):
            return (x < y)
        r = bigger(x, y)
        if r is not None:
            return not r
        return (x < y)

    def replue(x, y):
        b = bigger(x, y)
        if b is not None:
            return True
        return Unequality(x, y)

    def repl(ex, *args):
        if ex == S.true or ex == S.false:
            return bool(ex)
        return ex.replace(*args)
    expr = repl(expr, StrictLessThan, replie)
    expr = repl(expr, StrictGreaterThan, lambda x, y: replie(y, x))
    expr = repl(expr, Unequality, replue)
    return expr


@_noconds
def _laplace_transform(f, t, s_, simplify=True):
    """ The backend function for Laplace transforms. """
    from diofant import (re, Max, exp, pi, Min, periodic_argument as arg,
                       cos, Wild, symbols, polar_lift)
    s = Dummy('s')
    F = integrate(exp(-s*t) * f, (t, 0, oo))

    if not F.has(Integral):
        return _simplify(F.subs(s, s_), simplify), -oo, True

    if not F.is_Piecewise:
        raise IntegralTransformError(
            'Laplace', f, 'could not compute integral')

    F, cond = F.args[0]
    if F.has(Integral):
        raise IntegralTransformError(
            'Laplace', f, 'integral in unexpected form')

    def process_conds(conds):
        """ Turn ``conds`` into a strip and auxiliary conditions. """
        a = -oo
        aux = True
        conds = conjuncts(to_cnf(conds))
        u = Dummy('u', extended_real=True)
        p, q, w1, w2, w3, w4, w5 = symbols(
            'p q w1 w2 w3 w4 w5', cls=Wild, exclude=[s])
        for c in conds:
            a_ = oo
            aux_ = []
            for d in disjuncts(c):
                m = d.match(abs(arg((s + w3)**p*q, w1)) < w2)
                if not m:
                    m = d.match(abs(arg((s + w3)**p*q, w1)) <= w2)
                if not m:
                    m = d.match(abs(arg((polar_lift(s + w3))**p*q, w1)) < w2)
                if not m:
                    m = d.match(abs(arg((polar_lift(s + w3))**p*q, w1)) <= w2)
                if m:
                    if m[q].is_positive and m[w2]/m[p] == pi/2:
                        d = re(s + m[w3]) > 0
                m = d.match(
                    0 < cos(abs(arg(s**w1*w5, q))*w2)*abs(s**w3)**w4 - p)
                if not m:
                    m = d.match(0 < cos(abs(
                        arg(polar_lift(s)**w1*w5, q))*w2)*abs(s**w3)**w4 - p)
                if m and all(m[wild].is_positive for wild in [w1, w2, w3, w4, w5]):
                    d = re(s) > m[p]
                d_ = d.replace(
                    re, lambda x: x.expand().as_real_imag()[0]).subs(re(s), t)
                if not d.is_Relational or \
                    d.rel_op not in ('>', '>=', '<', '<=') \
                        or d_.has(s) or not d_.has(t):
                    aux_ += [d]
                    continue
                soln = solve_univariate_inequality(d_, t)
                t_ = Dummy("t", real=True)
                soln = soln.subs(t, t_).subs(t_, t)
                if not soln.is_Relational or \
                        soln.rel_op not in ('>', '>=', '<', '<='):
                    aux_ += [d]
                    continue
                if soln.lts == t:
                    raise IntegralTransformError('Laplace', f,
                                         'convergence not in half-plane?')
                else:
                    a_ = Min(soln.lts, a_)
            if a_ != oo:
                a = Max(a_, a)
            else:
                aux = And(aux, Or(*aux_))
        return a, aux

    conds = [process_conds(c) for c in disjuncts(cond)]
    conds2 = [x for x in conds if x[1] is not S.false and x[0] != -oo]
    if not conds2:
        conds2 = [x for x in conds if x[1] is not S.false]
    conds = conds2

    def cnt(expr):
        if expr == S.true or expr == S.false:
            return 0
        return expr.count_ops()
    conds.sort(key=lambda x: (-x[0], cnt(x[1])))

    if not conds:
        raise IntegralTransformError('Laplace', f, 'no convergence found')
    a, aux = conds[0]

    def sbs(expr):
        if expr == S.true or expr == S.false:
            return bool(expr)
        return expr.subs(s, s_)
    if simplify:
        F = _simplifyconds(F, s, a)
        aux = _simplifyconds(aux, s, a)
    return _simplify(F.subs(s, s_), simplify), sbs(a), sbs(aux)


class LaplaceTransform(IntegralTransform):
    """
    Class representing unevaluated Laplace transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute Laplace transforms, see the :func:`laplace_transform`
    docstring.
    """

    _name = 'Laplace'

    def _compute_transform(self, f, t, s, **hints):
        return _laplace_transform(f, t, s, **hints)

    def _as_integral(self, f, t, s):
        from diofant import exp
        return Integral(f*exp(-s*t), (t, 0, oo))

    def _collapse_extra(self, extra):
        from diofant import Max
        conds = []
        planes = []
        for plane, cond in extra:
            conds.append(cond)
            planes.append(plane)
        cond = And(*conds)
        plane = Max(*planes)
        if cond == S.false:
            raise IntegralTransformError(
                'Laplace', None, 'No combined convergence.')
        return plane, cond


def laplace_transform(f, t, s, **hints):
    r"""
    Compute the Laplace Transform `F(s)` of `f(t)`,

    .. math :: F(s) = \int_0^\infty e^{-st} f(t) \mathrm{d}t.

    For all "sensible" functions, this converges absolutely in a
    half plane  `a < \operatorname{Re}(s)`.

    This function returns ``(F, a, cond)``
    where ``F`` is the Laplace transform of ``f``, `\operatorname{Re}(s) > a` is the half-plane
    of convergence, and ``cond`` are auxiliary convergence conditions.

    If the integral cannot be computed in closed form, this function returns
    an unevaluated :class:`~diofant.integrals.transforms.LaplaceTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`. If ``noconds=True``,
    only `F` will be returned (i.e. not ``cond``, and also not the plane ``a``).

    >>> from diofant.integrals import laplace_transform
    >>> from diofant.abc import t, s, a
    >>> laplace_transform(t**a, t, s)
    (s**(-a)*gamma(a + 1)/s, 0, -re(a) < 1)

    See Also
    ========

    inverse_laplace_transform, mellin_transform, fourier_transform
    hankel_transform, inverse_hankel_transform
    """
    if isinstance(f, MatrixBase) and hasattr(f, 'applyfunc'):
        return f.applyfunc(lambda fij: laplace_transform(fij, t, s, **hints))
    return LaplaceTransform(f, t, s).doit(**hints)


@_noconds_(True)
def _inverse_laplace_transform(F, s, t_, plane, simplify=True):
    """ The backend function for inverse Laplace transforms. """
    from diofant import exp, Heaviside, log, expand_complex, Integral, Piecewise
    from diofant.integrals.meijerint import meijerint_inversion, _get_coeff_exp
    # There are two strategies we can try:
    # 1) Use inverse mellin transforms - related by a simple change of variables.
    # 2) Use the inversion integral.

    t = Dummy('t', extended_real=True)

    def pw_simp(*args):
        """ Simplify a piecewise expression from hyperexpand. """
        # XXX we break modularity here!
        if len(args) != 3:
            return Piecewise(*args)
        arg = args[2].args[0].argument
        coeff, exponent = _get_coeff_exp(arg, t)
        e1 = args[0].args[0]
        e2 = args[1].args[0]
        return Heaviside(1/abs(coeff) - t**exponent)*e1 \
            + Heaviside(t**exponent - 1/abs(coeff))*e2

    try:
        f, cond = inverse_mellin_transform(F, s, exp(-t), (None, oo),
                                           needeval=True, noconds=False)
    except IntegralTransformError:
        f = None
    if f is None:
        f = meijerint_inversion(F, s, t)
        if f is None:
            raise IntegralTransformError('Inverse Laplace', f, '')
        if f.is_Piecewise:
            f, cond = f.args[0]
            if f.has(Integral):
                raise IntegralTransformError('Inverse Laplace', f,
                                     'inversion integral of unrecognised form.')
        else:
            cond = True
        f = f.replace(Piecewise, pw_simp)

    if f.is_Piecewise:
        # many of the functions called below can't work with piecewise
        # (b/c it has a bool in args)
        return f.subs(t, t_), cond

    u = Dummy('u')

    def simp_heaviside(arg):
        a = arg.subs(exp(-t), u)
        if a.has(t):
            return Heaviside(arg)
        rel = solve_univariate_inequality(a > 0, u)
        u_ = Dummy('u', real=True)
        rel = rel.subs(u, u_).subs(u_, u)
        if rel.lts == u:
            k = log(rel.gts)
            return Heaviside(t + k)
        else:
            k = log(rel.lts)
            return Heaviside(-(t + k))
    f = f.replace(Heaviside, simp_heaviside)
    f = f.replace(lambda expr: expr.is_Pow and expr.base is S.Exp1,
                  lambda expr: expand_complex(exp(expr.exp)))

    # TODO it would be nice to fix cosh and sinh ... simplify messes these
    #      exponentials up

    return _simplify(f.subs(t, t_), simplify), cond


class InverseLaplaceTransform(IntegralTransform):
    """
    Class representing unevaluated inverse Laplace transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute inverse Laplace transforms, see the
    :func:`inverse_laplace_transform` docstring.
    """

    _name = 'Inverse Laplace'
    _none_sentinel = Dummy('None')
    _c = Dummy('c')

    def __new__(cls, F, s, x, plane, **opts):
        if plane is None:
            plane = InverseLaplaceTransform._none_sentinel
        return IntegralTransform.__new__(cls, F, s, x, plane, **opts)

    @property
    def fundamental_plane(self):
        plane = self.args[3]
        if plane is InverseLaplaceTransform._none_sentinel:
            plane = None
        return plane

    def _compute_transform(self, F, s, t, **hints):
        return _inverse_laplace_transform(F, s, t, self.fundamental_plane, **hints)

    def _as_integral(self, F, s, t):
        from diofant import I, exp
        c = self.__class__._c
        return Integral(exp(s*t)*F, (s, c - I*oo, c + I*oo))


def inverse_laplace_transform(F, s, t, plane=None, **hints):
    r"""
    Compute the inverse Laplace transform of `F(s)`, defined as

    .. math :: f(t) = \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s,

    for `c` so large that `F(s)` has no singularites in the
    half-plane `\operatorname{Re}(s) > c-\epsilon`.

    The plane can be specified by
    argument ``plane``, but will be inferred if passed as None.

    Under certain regularity conditions, this recovers `f(t)` from its
    Laplace Transform `F(s)`, for non-negative `t`, and vice
    versa.

    If the integral cannot be computed in closed form, this function returns
    an unevaluated :class:`~diofant.integrals.transforms.InverseLaplaceTransform` object.

    Note that this function will always assume `t` to be real,
    regardless of the diofant assumption on `t`.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.

    >>> from diofant.integrals.transforms import inverse_laplace_transform
    >>> from diofant import exp, Symbol
    >>> from diofant.abc import s, t
    >>> a = Symbol('a', positive=True)
    >>> inverse_laplace_transform(exp(-a*s)/s, s, t)
    Heaviside(-a + t)

    See Also
    ========

    laplace_transform
    hankel_transform, inverse_hankel_transform
    """
    if isinstance(F, MatrixBase) and hasattr(F, 'applyfunc'):
        return F.applyfunc(lambda Fij: inverse_laplace_transform(Fij, s, t, plane, **hints))
    return InverseLaplaceTransform(F, s, t, plane).doit(**hints)


##########################################################################
# Fourier Transform
##########################################################################

@_noconds_(True)
def _fourier_transform(f, x, k, a, b, name, simplify=True):
    """
    Compute a general Fourier-type transform
        F(k) = a int_-oo^oo exp(b*I*x*k) f(x) dx.

    For suitable choice of a and b, this reduces to the standard Fourier
    and inverse Fourier transforms.
    """
    from diofant import exp, I
    F = integrate(a*f*exp(b*I*x*k), (x, -oo, oo))

    if not F.has(Integral):
        return _simplify(F, simplify), True

    if not F.is_Piecewise:
        raise IntegralTransformError(name, f, 'could not compute integral')

    F, cond = F.args[0]
    if F.has(Integral):
        raise IntegralTransformError(name, f, 'integral in unexpected form')

    return _simplify(F, simplify), cond


class FourierTypeTransform(IntegralTransform):
    """ Base class for Fourier transforms."""

    def a(self):
        raise NotImplementedError(
            "Class %s must implement a(self) but does not" % self.__class__)

    def b(self):
        raise NotImplementedError(
            "Class %s must implement b(self) but does not" % self.__class__)

    def _compute_transform(self, f, x, k, **hints):
        return _fourier_transform(f, x, k,
                                  self.a(), self.b(),
                                  self.__class__._name, **hints)

    def _as_integral(self, f, x, k):
        from diofant import exp, I
        a = self.a()
        b = self.b()
        return Integral(a*f*exp(b*I*x*k), (x, -oo, oo))


class FourierTransform(FourierTypeTransform):
    """
    Class representing unevaluated Fourier transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute Fourier transforms, see the :func:`fourier_transform`
    docstring.
    """

    _name = 'Fourier'

    def a(self):
        return 1

    def b(self):
        return -2*S.Pi


def fourier_transform(f, x, k, **hints):
    r"""
    Compute the unitary, ordinary-frequency Fourier transform of `f`, defined
    as

    .. math:: F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.FourierTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import fourier_transform, exp
    >>> from diofant.abc import x, k
    >>> fourier_transform(exp(-x**2), x, k)
    E**(-pi**2*k**2)*sqrt(pi)
    >>> fourier_transform(exp(-x**2), x, k, noconds=False)
    (E**(-pi**2*k**2)*sqrt(pi), True)

    See Also
    ========

    inverse_fourier_transform
    sine_transform, inverse_sine_transform
    cosine_transform, inverse_cosine_transform
    hankel_transform, inverse_hankel_transform
    mellin_transform, laplace_transform
    """
    return FourierTransform(f, x, k).doit(**hints)


class InverseFourierTransform(FourierTypeTransform):
    """
    Class representing unevaluated inverse Fourier transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute inverse Fourier transforms, see the
    :func:`inverse_fourier_transform` docstring.
    """

    _name = 'Inverse Fourier'

    def a(self):
        return 1

    def b(self):
        return 2*S.Pi


def inverse_fourier_transform(F, k, x, **hints):
    r"""
    Compute the unitary, ordinary-frequency inverse Fourier transform of `F`,
    defined as

    .. math:: f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.InverseFourierTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import inverse_fourier_transform, exp, sqrt, pi
    >>> from diofant.abc import x, k
    >>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x)
    E**(-x**2)
    >>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False)
    (E**(-x**2), True)

    See Also
    ========

    fourier_transform
    sine_transform, inverse_sine_transform
    cosine_transform, inverse_cosine_transform
    hankel_transform, inverse_hankel_transform
    mellin_transform, laplace_transform
    """
    return InverseFourierTransform(F, k, x).doit(**hints)


##########################################################################
# Fourier Sine and Cosine Transform
##########################################################################

from diofant import sin, cos, sqrt, pi


@_noconds_(True)
def _sine_cosine_transform(f, x, k, a, b, K, name, simplify=True):
    """
    Compute a general sine or cosine-type transform
        F(k) = a int_0^oo b*sin(x*k) f(x) dx.
        F(k) = a int_0^oo b*cos(x*k) f(x) dx.

    For suitable choice of a and b, this reduces to the standard sine/cosine
    and inverse sine/cosine transforms.
    """
    F = integrate(a*f*K(b*x*k), (x, 0, oo))

    if not F.has(Integral):
        return _simplify(F, simplify), True

    if not F.is_Piecewise:
        raise IntegralTransformError(name, f, 'could not compute integral')

    F, cond = F.args[0]
    if F.has(Integral):
        raise IntegralTransformError(name, f, 'integral in unexpected form')

    return _simplify(F, simplify), cond


class SineCosineTypeTransform(IntegralTransform):
    """
    Base class for sine and cosine transforms.
    Specify cls._kern.
    """

    def a(self):
        raise NotImplementedError(
            "Class %s must implement a(self) but does not" % self.__class__)

    def b(self):
        raise NotImplementedError(
            "Class %s must implement b(self) but does not" % self.__class__)

    def _compute_transform(self, f, x, k, **hints):
        return _sine_cosine_transform(f, x, k,
                                      self.a(), self.b(),
                                      self.__class__._kern,
                                      self.__class__._name, **hints)

    def _as_integral(self, f, x, k):
        a = self.a()
        b = self.b()
        K = self.__class__._kern
        return Integral(a*f*K(b*x*k), (x, 0, oo))


class SineTransform(SineCosineTypeTransform):
    """
    Class representing unevaluated sine transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute sine transforms, see the :func:`sine_transform`
    docstring.
    """

    _name = 'Sine'
    _kern = sin

    def a(self):
        return sqrt(2)/sqrt(pi)

    def b(self):
        return 1


def sine_transform(f, x, k, **hints):
    r"""
    Compute the unitary, ordinary-frequency sine transform of `f`, defined
    as

    .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.SineTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import sine_transform, exp
    >>> from diofant.abc import x, k, a
    >>> sine_transform(x*exp(-a*x**2), x, k)
    sqrt(2)*E**(-k**2/(4*a))*k/(4*a**(3/2))
    >>> sine_transform(x**(-a), x, k)
    2**(-a + 1/2)*k**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + 1/2)

    See Also
    ========

    fourier_transform, inverse_fourier_transform
    inverse_sine_transform
    cosine_transform, inverse_cosine_transform
    hankel_transform, inverse_hankel_transform
    mellin_transform, laplace_transform
    """
    return SineTransform(f, x, k).doit(**hints)


class InverseSineTransform(SineCosineTypeTransform):
    """
    Class representing unevaluated inverse sine transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute inverse sine transforms, see the
    :func:`inverse_sine_transform` docstring.
    """

    _name = 'Inverse Sine'
    _kern = sin

    def a(self):
        return sqrt(2)/sqrt(pi)

    def b(self):
        return 1


def inverse_sine_transform(F, k, x, **hints):
    r"""
    Compute the unitary, ordinary-frequency inverse sine transform of `F`,
    defined as

    .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.InverseSineTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import inverse_sine_transform, exp, sqrt, gamma, pi
    >>> from diofant.abc import x, k, a
    >>> inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)*
    ...     gamma(-a/2 + 1)/gamma((a+1)/2), k, x)
    x**(-a)
    >>> inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x)
    E**(-a*x**2)*x

    See Also
    ========

    fourier_transform, inverse_fourier_transform
    sine_transform
    cosine_transform, inverse_cosine_transform
    hankel_transform, inverse_hankel_transform
    mellin_transform, laplace_transform
    """
    return InverseSineTransform(F, k, x).doit(**hints)


class CosineTransform(SineCosineTypeTransform):
    """
    Class representing unevaluated cosine transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute cosine transforms, see the :func:`cosine_transform`
    docstring.
    """

    _name = 'Cosine'
    _kern = cos

    def a(self):
        return sqrt(2)/sqrt(pi)

    def b(self):
        return 1


def cosine_transform(f, x, k, **hints):
    r"""
    Compute the unitary, ordinary-frequency cosine transform of `f`, defined
    as

    .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.CosineTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import cosine_transform, exp, sqrt, cos
    >>> from diofant.abc import x, k, a
    >>> cosine_transform(exp(-a*x), x, k)
    sqrt(2)*a/(sqrt(pi)*(a**2 + k**2))
    >>> cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k)
    E**(-a**2/(2*k))*a/(2*k**(3/2))

    See Also
    ========

    fourier_transform, inverse_fourier_transform,
    sine_transform, inverse_sine_transform
    inverse_cosine_transform
    hankel_transform, inverse_hankel_transform
    mellin_transform, laplace_transform
    """
    return CosineTransform(f, x, k).doit(**hints)


class InverseCosineTransform(SineCosineTypeTransform):
    """
    Class representing unevaluated inverse cosine transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute inverse cosine transforms, see the
    :func:`inverse_cosine_transform` docstring.
    """

    _name = 'Inverse Cosine'
    _kern = cos

    def a(self):
        return sqrt(2)/sqrt(pi)

    def b(self):
        return 1


def inverse_cosine_transform(F, k, x, **hints):
    r"""
    Compute the unitary, ordinary-frequency inverse cosine transform of `F`,
    defined as

    .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.InverseCosineTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import inverse_cosine_transform, exp, sqrt, pi
    >>> from diofant.abc import x, k, a
    >>> inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x)
    E**(-a*x)
    >>> inverse_cosine_transform(1/sqrt(k), k, x)
    1/sqrt(x)

    See Also
    ========

    fourier_transform, inverse_fourier_transform,
    sine_transform, inverse_sine_transform
    cosine_transform
    hankel_transform, inverse_hankel_transform
    mellin_transform, laplace_transform
    """
    return InverseCosineTransform(F, k, x).doit(**hints)


##########################################################################
# Hankel Transform
##########################################################################

@_noconds_(True)
def _hankel_transform(f, r, k, nu, name, simplify=True):
    """
    Compute a general Hankel transform

    .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.
    """
    from diofant import besselj
    F = integrate(f*besselj(nu, k*r)*r, (r, 0, oo))

    if not F.has(Integral):
        return _simplify(F, simplify), True

    if not F.is_Piecewise:
        raise IntegralTransformError(name, f, 'could not compute integral')

    F, cond = F.args[0]
    if F.has(Integral):
        raise IntegralTransformError(name, f, 'integral in unexpected form')

    return _simplify(F, simplify), cond


class HankelTypeTransform(IntegralTransform):
    """
    Base class for Hankel transforms.
    """

    def doit(self, **hints):
        return self._compute_transform(self.function,
                                       self.function_variable,
                                       self.transform_variable,
                                       self.args[3],
                                       **hints)

    def _compute_transform(self, f, r, k, nu, **hints):
        return _hankel_transform(f, r, k, nu, self._name, **hints)

    def _as_integral(self, f, r, k, nu):
        from diofant import besselj
        return Integral(f*besselj(nu, k*r)*r, (r, 0, oo))

    @property
    def as_integral(self):
        return self._as_integral(self.function,
                                 self.function_variable,
                                 self.transform_variable,
                                 self.args[3])


class HankelTransform(HankelTypeTransform):
    """
    Class representing unevaluated Hankel transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute Hankel transforms, see the :func:`hankel_transform`
    docstring.
    """

    _name = 'Hankel'


def hankel_transform(f, r, k, nu, **hints):
    r"""
    Compute the Hankel transform of `f`, defined as

    .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.HankelTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import hankel_transform, inverse_hankel_transform
    >>> from diofant import gamma, exp, sinh, cosh
    >>> from diofant.abc import r, k, m, nu, a

    >>> ht = hankel_transform(1/r**m, r, k, nu)
    >>> ht
    2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)

    >>> inverse_hankel_transform(ht, k, r, nu)
    r**(-m)

    >>> ht = hankel_transform(exp(-a*r), r, k, 0)
    >>> ht
    a/(k**3*(a**2/k**2 + 1)**(3/2))

    >>> inverse_hankel_transform(ht, k, r, 0)
    E**(-a*r)

    See Also
    ========

    fourier_transform, inverse_fourier_transform
    sine_transform, inverse_sine_transform
    cosine_transform, inverse_cosine_transform
    inverse_hankel_transform
    mellin_transform, laplace_transform
    """
    return HankelTransform(f, r, k, nu).doit(**hints)


class InverseHankelTransform(HankelTypeTransform):
    """
    Class representing unevaluated inverse Hankel transforms.

    For usage of this class, see the :class:`~diofant.integrals.transforms.IntegralTransform` docstring.

    For how to compute inverse Hankel transforms, see the
    :func:`inverse_hankel_transform` docstring.
    """

    _name = 'Inverse Hankel'


def inverse_hankel_transform(F, k, r, nu, **hints):
    r"""
    Compute the inverse Hankel transform of `F` defined as

    .. math:: f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k.

    If the transform cannot be computed in closed form, this
    function returns an unevaluated :class:`~diofant.integrals.transforms.InverseHankelTransform` object.

    For a description of possible hints, refer to the docstring of
    :func:`diofant.integrals.transforms.IntegralTransform.doit`.
    Note that for this transform, by default ``noconds=True``.

    >>> from diofant import hankel_transform, inverse_hankel_transform, gamma
    >>> from diofant import gamma, exp, sinh, cosh
    >>> from diofant.abc import r, k, m, nu, a

    >>> ht = hankel_transform(1/r**m, r, k, nu)
    >>> ht
    2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)

    >>> inverse_hankel_transform(ht, k, r, nu)
    r**(-m)

    >>> ht = hankel_transform(exp(-a*r), r, k, 0)
    >>> ht
    a/(k**3*(a**2/k**2 + 1)**(3/2))

    >>> inverse_hankel_transform(ht, k, r, 0)
    E**(-a*r)

    See Also
    ========

    fourier_transform, inverse_fourier_transform
    sine_transform, inverse_sine_transform
    cosine_transform, inverse_cosine_transform
    hankel_transform
    mellin_transform, laplace_transform
    """
    return InverseHankelTransform(F, k, r, nu).doit(**hints)
