Consider the (possibly singular or hypersingular) integral equation $$Z\int_{-\infty}^\infty B(x,x^\prime) \ A(x^\prime) dx^\prime = A(x)$$ where $Z$ indicates the finite part of the integral should be taken.
I have been thinking about both $x$ and $x^\prime$ as real numbers. Is there any problem with replacing $x$ with a complex variable $z$ and considering the same equation as also defining the analytic continuation of $A(x)$ into the complex plane? Are there any pitfalls I should watch out for where this simple approach becomes ill-defined?
For concreteness, the specific example I am considering is: $$B(x,x^\prime) = \frac{\beta a}{\pi \sqrt{2}} \bigg( \frac{K_1(\beta \left|x-x^\prime\right|)}{\sqrt{2} \left|x - x^\prime\right|} + \frac{K_1(\beta \sqrt{x^2+x^{\prime^2}})}{\sqrt{2(x^2+x^{\prime^{\ 2}})}}\bigg)$$ where $\beta$ and $a$ are constants.