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from sympy.functions import SingularityFunction, DiracDelta
from sympy.integrals import integrate
def singularityintegrate(f, x):
"""
This function handles the indefinite integrations of Singularity functions.
The ``integrate`` function calls this function internally whenever an
instance of SingularityFunction is passed as argument.
Explanation
===========
The idea for integration is the following:
- If we are dealing with a SingularityFunction expression,
i.e. ``SingularityFunction(x, a, n)``, we just return
``SingularityFunction(x, a, n + 1)/(n + 1)`` if ``n >= 0`` and
``SingularityFunction(x, a, n + 1)`` if ``n < 0``.
- If the node is a multiplication or power node having a
SingularityFunction term we rewrite the whole expression in terms of
Heaviside and DiracDelta and then integrate the output. Lastly, we
rewrite the output of integration back in terms of SingularityFunction.
- If none of the above case arises, we return None.
Examples
========
>>> from sympy.integrals.singularityfunctions import singularityintegrate
>>> from sympy import SingularityFunction, symbols, Function
>>> x, a, n, y = symbols('x a n y')
>>> f = Function('f')
>>> singularityintegrate(SingularityFunction(x, a, 3), x)
SingularityFunction(x, a, 4)/4
>>> singularityintegrate(5*SingularityFunction(x, 5, -2), x)
5*SingularityFunction(x, 5, -1)
>>> singularityintegrate(6*SingularityFunction(x, 5, -1), x)
6*SingularityFunction(x, 5, 0)
>>> singularityintegrate(x*SingularityFunction(x, 0, -1), x)
0
>>> singularityintegrate(SingularityFunction(x, 1, -1) * f(x), x)
f(1)*SingularityFunction(x, 1, 0)
"""
if not f.has(SingularityFunction):
return None
if isinstance(f, SingularityFunction):
x, a, n = f.args
if n.is_positive or n.is_zero:
return SingularityFunction(x, a, n + 1)/(n + 1)
elif n in (-1, -2, -3, -4):
return SingularityFunction(x, a, n + 1)
if f.is_Mul or f.is_Pow:
expr = f.rewrite(DiracDelta)
expr = integrate(expr, x)
return expr.rewrite(SingularityFunction)
return None
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