Suppose you got a function $f(x)$ with a singularity $s$, point $a$ and a small number $\epsilon$.
For what $b$ does this equation hold? $$\int_{s-a}^{s-\epsilon}f(x) dx + \int_{s+\epsilon}^{s+b}f(x) dx = 0$$ I want to evaluate $b$ to at least 12 digits of precision.
Example:
$f(x)=\psi(x)$, $s=-2$, $a=10^{-5}$, $\epsilon=10^{-12}$
$\psi(x)$ is the Digamma function, which has a singularity when $x=-2$. For what $b$ does this equation hold? $$\int_{ -2-10^{-5}}^{-2-10^{-12}}\psi(x) dx + \int_{-2+10^{-12}}^{-2+b}\psi(x) dx = 0$$
