Comment #13 on issue 2723 by asmeurer: What should summation() do with
non-integer limits?
http://code.google.com/p/sympy/issues/detail?id=2723
So there are at least three different definitions of summations with
non-integer limits a and b:
- Only sum over the integers in the range [a, b] (the Maple version)
- Sum f(a), f(a + 1), ..., f(a + n) where n = floor(b - a)
- Compute the indefinite summation and substitute values.
I believe the second option and third option agree (except possibly at some
removable singularities) because of the way that indefinite summations are
defined with difference operators (though I'm not certain about it).
There is actually a fourth option, which is a mix of the first and second,
where you offset the initial point by 1, but then only consider integers in
that range. This appears to be what Mathematica does. See for example
http://www.wolframalpha.com/input/?i=sum(exp(a*x)%2C+x%3D-1%2F2..1%2F4).
I'm curious if there is a system that would allow 1/2 step instead of the
default 1.
I doubt this is formalized, as you can get the exact same effect simply by
dividing the index by 2 in the summand.
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