deltafunctions.py

from ..core import Mul
from ..functions import DiracDelta, Heaviside
from ..utilities import default_sort_key


def change_mul(node, x):
    """change_mul(node, x)

    Rearranges the operands of a product, bringing to front any simple
    DiracDelta expression.

    If no simple DiracDelta expression was found, then all the DiracDelta
    expressions are simplified (using DiracDelta.simplify).

    Return: (dirac, new node)
    Where:
        o dirac is either a simple DiracDelta expression or None (if no simple
          expression was found);
        o new node is either a simplified DiracDelta expressions or None (if it
          could not be simplified).

    Examples
    ========

    >>> change_mul(x*y*DiracDelta(x)*cos(x), x)
    (DiracDelta(x), x*y*cos(x))
    >>> change_mul(x*y*DiracDelta(x**2 - 1)*cos(x), x)
    (None, x*y*cos(x)*DiracDelta(x - 1)/2 + x*y*cos(x)*DiracDelta(x + 1)/2)
    >>> change_mul(x*y*DiracDelta(cos(x))*cos(x), x)
    (None, None)

    See Also
    ========

    diofant.functions.special.delta_functions.DiracDelta
    deltaintegrate

    """
    if not (node.is_Mul or node.is_Pow):
        return node

    new_args = []
    dirac = None

    # Sorting is needed so that we consistently collapse the same delta;
    # However, we must preserve the ordering of non-commutative terms
    c, nc = node.args_cnc()
    sorted_args = sorted(c, key=default_sort_key)
    sorted_args.extend(nc)

    for arg in sorted_args:
        if arg.is_Pow and isinstance(arg.base, DiracDelta):
            new_args.append(arg.func(arg.base, arg.exp - 1))
            arg = arg.base
        if dirac is None and (isinstance(arg, DiracDelta) and arg.is_simple(x)
                              and (len(arg.args) <= 1 or arg.args[1] == 0)):
            dirac = arg
        else:
            new_args.append(arg)
    if not dirac:  # there was no simple dirac
        new_args = []
        for arg in sorted_args:
            if isinstance(arg, DiracDelta):
                new_args.append(arg.simplify(x))
            elif arg.is_Pow and isinstance(arg.base, DiracDelta):
                new_args.append(arg.func(arg.base.simplify(x), arg.exp))
            else:
                new_args.append(change_mul(arg, x))
        if new_args != sorted_args:
            nnode = Mul(*new_args).expand()
        else:  # if the node didn't change there is nothing to do
            nnode = None
        return None, nnode
    return dirac, Mul(*new_args)


def deltaintegrate(f, x):
    """
    deltaintegrate(f, x)

    The idea for integration is the following:

    - If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)),
      we try to simplify it.

      If we could simplify it, then we integrate the resulting expression.
      We already know we can integrate a simplified expression, because only
      simple DiracDelta expressions are involved.

      If we couldn't simplify it, there are two cases:

      1) The expression is a simple expression: we return the integral,
         taking care if we are dealing with a Derivative or with a proper
         DiracDelta.

      2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do
         nothing at all.

    - If the node is a multiplication node having a DiracDelta term:

      First we expand it.

      If the expansion did work, then we try to integrate the expansion.

      If not, we try to extract a simple DiracDelta term, then we have two
      cases:

      1) We have a simple DiracDelta term, so we return the integral.

      2) We didn't have a simple term, but we do have an expression with
         simplified DiracDelta terms, so we integrate this expression.

    Examples
    ========

        >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x)
        sin(1)*cos(1)*Heaviside(x - 1)
        >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y)
        z**2*DiracDelta(x - z)*Heaviside(y - z)

    See Also
    ========

    diofant.functions.special.delta_functions.DiracDelta
    diofant.integrals.integrals.Integral

    """
    if not f.has(DiracDelta):
        return

    from ..solvers import solve
    from .integrals import Integral, integrate

    # g(x) = DiracDelta(h(x))
    if f.func == DiracDelta:
        h = f.simplify(x)
        if h == f:  # can't simplify the expression
            # FIXME: the second term tells whether is DeltaDirac or Derivative
            # For integrating derivatives of DiracDelta we need the chain rule
            if f.is_simple(x):
                if (len(f.args) <= 1 or f.args[1] == 0):
                    return Heaviside(f.args[0])
                else:
                    return (DiracDelta(f.args[0], f.args[1] - 1) /
                            f.args[0].as_poly().LC())
        else:  # let's try to integrate the simplified expression
            fh = integrate(h, x)
            return fh
    elif f.is_Mul or f.is_Pow:  # g(x) = a*b*c*f(DiracDelta(h(x)))*d*e
        g = f.expand()
        if f != g:  # the expansion worked
            fh = integrate(g, x)
            if fh is not None and not isinstance(fh, Integral):
                return fh
        else:
            # no expansion performed, try to extract a simple DiracDelta term
            dg, rest_mult = change_mul(f, x)

            if not dg:
                if rest_mult:
                    fh = integrate(rest_mult, x)
                    return fh
            else:
                dg = dg.simplify(x)
                if dg.is_Mul:  # Take out any extracted factors
                    dg, rest_mult_2 = change_mul(dg, x)
                    rest_mult = rest_mult*rest_mult_2
                point = solve(dg.args[0], x)[0][x]
                return (rest_mult.subs({x: point})*Heaviside(x - point))