My question here is distinct to those that appear to have been asked earlier here, such as 19833.
The broad problem is that Mathematica is not simplifying the Assumptions given before usage. This is the simplest example similar to where the problem is occuring in my work, which is basically the orthogonality of the Spherical Harmonics with
This gives an output of
0. Now, the assumptions I've given already constrain
m==0, in which case I should have a nonzero output. In fact
Reduce[m∈Integers&&-1<m<1] gives me
m==0, as expected, and then
gives me the expected answer of
So the question is basically this: Why isn't Mathematica giving the same answer under two equivalent set of Assumptions?
Also, in reference to the discussions in other questions like 19833, I'd like to point out that this equivalent behavior is obtained if I put the Assumptions inside the Integrate function instead, i.e., it still gives 0 when the assumption looks like
m∈Integers&&-1<m<1, and the nonzero answer when the assumption looks like
EDIT: Why I think this question is distinct from the previously asked ones: My understanding of the resolution to the previously asked questions would indicate that using
Integrate[...,Assumptions->...] would resolve the issue, which in this case it does not, as explicitly mentioned above. Further, from previous answers I would not be inclined to believe that
Assuming[m∈Integers&&-1<m<1, Integrate[...] and
Assuming[Reduce[m∈Integers&&-1<m<1], Integrate[...] would give distinct results, which it does in this case.