# What kind of numerical methods are best applicable to this?

I'm wondering: what would be the best numerical method for solving a nonlinear integral equation of the form

$$f(x) = a(x) + \int_{-A}^{A} K(x, t, f(t)) dt$$

where $f$ is the unknown function, a complex-valued function of the real variable $x$, $a$ is a known complex-valued function of $x$, and $K(x, t, z)$ is a complex-valued function of the real variables $x$ and $t$ and the complex variable $z$, and is analytic in $z$?

In particular, I'm interested in the case with $K(x, t, z) = \frac{1}{2\pi} \left(\frac{\exp(z)}{1 + it + ix} - \frac{\log(z)}{-1 + it - ix}\right)$ and $a(x) = \frac{L}{2\pi i}\left(\log(-1 + iA - ix) - \log(1 + iA - ix)\right) + \frac{\bar{L}}{2 \pi i}\left(\log(1 - iA - ix) - \log(-1 - iA - ix)\right)$ with $L = -W(-1)$ (Lambert $W$) and the second pair of logs is continuous "from below" at their cut on $(-\infty, 0]$.

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