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