I came across another interesting integral equations problem:
Consider the integral equation (inspired by non-relativistic scattering of an electron):
$\phi(x)=exp(ikx)+\int_{\mathbb{R}}\frac{exp(ik|x-t|)}{2ik}V(t)\phi(t)$, where $\phi(x)$ represents the wave function of the single particle.
I am assuming $V,\phi$ are $L^2$ functions and $k$ is complex, $V$ is a known function
PROBLEM:
I want to compute the Taylor series for $D(\lambda)=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}c_n\lambda_ n$
and ${D(x,t;\lambda)}=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}B_n(x,t)\lambda_n$
where $B_0(x,t)=K(x,t)$, $B_n(x,t)=c_nK(x,t)-n\int_{a}^{b}K(x,s)B_{n-1}(s,t)ds$
$c_n=\int_{a}^{b}B_n(t,t)dt$, with $c_0=1$.
WORK
I tried tracing through the formula, but this doesn't work, as the formula is for finite domains.