 Imaginary-part regularization at a zero Mellin exponent

Let $a>0$ and $t>0$. Near zero, the complex integral ${\int }_0{x}^{-1}{e}^{-tx}/(ia+\log x)dx$ fails both absolute and ordinary convergence through its real part, which behaves as $dx/(x\log x)$. Taking the imaginary part first givesThe regularized integrand $e^{-tx}/[x(a^2+{\log }^{2}x)]$ is integrable at zero: setting $u=-\log x$ changes its tail to a function comparable to $du/u^2$. Exponential decay controls infinity, and $a>0$ prevents an interior singularity at $x=1$.

A family with positive Mellin transform exponent can consequently have a finite imaginary-part boundary limit even when its full complex limit does not exist. Taking the limit of the full divergent expression and only then selecting its imaginary part is not justified. Endpoint-expansion coefficients can also change nonuniformly: a sine zero may cancel a gamma function pole and leave a nonzero constant term. This explains the regularization needed in the logarithmic expansion of Ramanujan's function.