By definition (e.g., https://en.wikipedia.org/wiki/Dirac_delta_function#As_a_measure), DiracDelta would have integral one over any set that contains 0, even if that set has only one point. So one has to ask, when we integrate over (x, a, b), do we mean closed interval [a, b], open (a, b), or half-open [a, b).
With either of two former choices we run into trouble with additivity over intervals: the integral of DiracDelta over [-1, 0] plus its integral over [0, 1] is 2, while the integral over [-1, 1] is 1. Similarly for open. (SymPy currently sidesteps this issue by returning answers in terms of Heaviside(0) which is left undefined.) So I think we should treat the intervals of integration as half-open: a <= x < b. Then an interval such as (0, 0) is empty set, and the integral of anything over an empty set is zero. I would also say that things like Piecewise with DiracDelta inside are not necessarily well-defined. Since DiracDelta is not a function, one should not define an integrand as "something when x != 3, and DiracDelta when x is 3". By implementing DiracDelta as a function, SymPy allows some expressions, like DiracDelta(x)**2, which have no mathematical meaning; and then the output is meaningless too. >>> integrate(DiracDelta(x)**2, (x, -1, 1)) DiracDelta(0) # garbage in, garbage out -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at https://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/CAJ0_tQJ0R1CkfAdYRD4Wh1mLUpYCQ_sb3xiiFeivfA--zDtYdw%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
