Imaginary-part regularization at a zero Mellin exponent ($\mathrm{Im}\left[-\frac{1}{a}\frac{1}{ia+\log x}\right]=\frac{1}{a^2+{\log }^{2}x}$)

At exponent zero the complex integral ${\int }_0{x}^{-1}{e}^{-tx}/(ia+\log x)dx$ fails absolute and ordinary convergence through its real part, proportional to $dx/(x\log x)$. The imaginary-part kernel displayed above instead decays as $dx/(x{\log }^{2}x)$ and is integrable. Thus a positive-exponent family can have a well-defined imaginary-part boundary limit even when its full complex limit is invalid. Endpoint expansion coefficients need not be uniform in that limiting exponent: a sine zero can cancel a gamma-function pole and introduce a new nonzero constant term.