Consider the Fredholm equation of the 2nd kind
$$ f(s) = \lambda \int_{-\infty}^{\infty} f(s') \Big(\sum_{n=1}^{N} g_n(s) h_n(s') \Big) ds' , $$
with $f(s)$ an unknown function, $\lambda$ a constant, {$h_1(s') , . . . , h_N(s')$} and {$g_1(s) , . . . , g_N(s)$} two sets of known functions.
Please note that this integral must be interpreted in the Principal Value sense. If the sets {$g_n(s)$}, {$h_n(s')$} contain only functions which decay sufficiently fast and contain no poles on the interval $(-\infty,\infty)$, then methods to solve this equation are covered in most introductory discussions of integral equations. However, what can be made of the case where at least one element of {$g_n(s)$} has at least one simple pole on the real axis?
Questions:
(1) How can this equation be classified? Is it a singular integral equation even though it does not involve poles 'along the diagonal'?
(2) Is there an accepted reference for integral equations of this type?
Note: If it helps, I am interested, to start, in a specific example: $$ f(s) = \frac{1}{a \beta} \int_{-\infty}^{\infty} f(s') \frac{s \sinh(\frac{\pi s}{2})}{s \sinh(\frac{\pi s}{2}) - 1} \frac{1}{\cosh(\frac{\pi (s-s')}{2})} ds' , $$ which has simple poles at $ s = \pm s_0 \approx \pm .7202 $. I know, from the solution of a transformed version of this equation that in $\lim_{a->\pm \infty} f(s)$ is proportional to a sum of Dirac deltas. So, clearly there is something pathological going on because the solution is not a function at all. It is not even clear to me that we may legitimately expand the kernel so that it follows the form that I have listed above.
I may need to solve several equations of this form, so I am interested in a general solution method.
