If you are taking the limit of an integral, can you switch the composition ie: take the integral of a limit if the limit and integral are on seperate variables? Ie: Lim of z to a [integral over alpha [f(x)/((x-z)(x-a)^2) dx] = Integral over alpha[Lim of z to a[f(x)/((x-z)(x-a)^2) dx] = Integral over alpha[f(x)/(x-a)^3 dx]
Since an integration involves limits, and switching the order of taking limits is a tricky business, i'd say that you could do the permutation, as long as there's no infinity (or no indeterminate expression under the limit sign) involved...My guess...
Generally speaking, if both limit and integral are "uniformly convergent" then they can be interchanged.