"duc " <firstname.lastname@example.org> wrote in message <email@example.com>...
> i need to integrate the following indefinite function
> exp((-(x-y)^2)/2*y^2)/sqrt(2*pi*y^2) ws.r.t y
> As it has an essential singularity i understand i need to integrate it numerically but i'm not sure which method will work, and if so which one is most effective in dealing with such functions
> your help would be much appreciated
- - - - - - - -
When you write -(x-y)^2)/2*y^2, that puts the y^2 in the numerator. Is that what you mean or do you mean -(x-y)^2)/(2*y^2) with the y^2 in the denominator? I am guessing you actually mean to have y^2 in the denominator.
With that supposition, let us assume that x > 0 and define a change of variable:
There is a pole singularity in this latter integral as the lower limit, (x-y)/y, approaches -1 but for any (x-y)/y > -1, it is convergent. This shows that the original integral in y based at y = 0 is well-defined for finite y >= 0 but diverges as y approaches plus infinity.
The above transformation to t not only indicates the nature of the singularity involved but also enables one to express the integral as a function of the single variable (x-y)/y as a lower limit without an explicit x being present in the integrand. Any of the standard quadrature routines can accomplish an evaluation of the above integral in t numerically with the proviso that the upper limit must stop short of infinity but be large enough to provide the necessary accuracy. You cannot of course start at t = -1 as a lower limit.
With x or y in the negative range, the situation is similar to the above.
If my guess above in the first paragraph was wrong, your integral will be divergent as y approaches zero and you will need to base it somewhere else.
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