I am trying to find a closed-form expression for the following integral:
with
Neither Maple nor Mathematica will give a ready answer.
If I put to 0, Maple gives me:
With Ei the exponential integral. This is Ok as far as it goes, but I need the integral for .
If I add to the function, Maple can no longer find the integral.
I tried taking the derivative of for clues, and that gives me , and the factor is not what I'm looking for; I need .
Any suggestions? Am I overlooking something obvious?
Well, the integral arises as the expectation of 1/x of a statistic with a truncated Normal distribution. If you use an ordinary non-truncated Normal distribution, the distribution of 1/x is the Cauchy distribution, which doesn't have an expectation. But I don't think that helps, and the answer seems to revolve around special functions anyway so I didn't post it in the probability and statistics thread.
If I allow I have to keep track of the singularity of in 0, which I'm not sure how to do. This work has already been done for the case that in the investigation of the Ei function,which isn't something I know how to replicate.
So why not define a special function like e.g. and go with that?
Well, for two reasons.
First of all, the integral seems tantalizingly close to one that's already expressible in terms of a known special function (Ei). If I'm to define a "new" special function, I should have some confidence I'm not overlooking some obvious trick that will reduce it to a known special function, right. I have no idea how to go about that. Any suggestions on this part?
If I can be reasonable certain that my integral isn't expressible in known elementary and special functions I'm happy to define a new special function as you suggest and do a simple investigation of its properties (extremes, inflexion points, limits).
Secondly, I can get numerical values for the integral I'm looking for from Matlab, but that just gives me numbers but not a lot of insight. Besides I can't readily check the numbers if I do, so I'd either have to trust Maple to get it right, or program the function in a dedicated numerical program like Matlab, Scilab or Octave and then use a numerical integration algorithm that I know to be correct to check on Maple's numbers. I will if I have to, but again, I want to be sure that it's not redundant (i.e. that isn't some simple expression of documented quantities).
Of course, which is why I'm interested in the truncated Normal (so that I can exclude 0 from the integration domain). In the question of the integral as posed the problem of the singularity for plays no role.
The digression about the singularity and the expectation of was by way of background information, which I left out because I felt it might obscure the real problem,which is how to deal with the integral.