Source code for diofant.integrals.deltafunctions

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


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)


[docs]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 .integrals import Integral, integrate from ..solvers import solve # 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)) return