Resolvent kernel

Summary

The resolvent kernel $R(x, s; \lambda)$ of a kernel $K$ gives the solution of the integral equation of the second kind $\varphi = f + \lambda K \varphi$. The solution is $\varphi(x) = f(x) + \lambda \int_a^b R(x, s; \lambda) f(s) \, ds$. For small $\abs{\lambda}$ it is the sum $\sum_{n \ge 1} \lambda^{n-1} K_n$ of the iterated kernels. It thus inverts the operator $I - \lambda K$ once and for all, for every right side $f$. For a continuous kernel it extends to a meromorphic function of $\lambda$, whose poles are characteristic values of $K$.