= Imaginary-part regularization at a zero Mellin exponent
{title2=$\operatorname{Im}\!\left[-\frac1a\frac1{ia+\log x}\right]=\frac1{a^2+\log^2x}$}
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 $dx/(x\log^2x)$ 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.
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