= Parameter-shift remainder bound for a logarithmic Laplace integral
{title2=$|g(s;t,a,\sigma)|\le\Gamma(s+\sigma)/(a^2t^{s+\sigma})$}
For $g(s)=-(1/a)e^{ias}\int_0^\infty x^{s+\sigma-1}e^{-tx}/(ia+\log x)\,dx$ with positive $a,t,\sigma$ and $s\ge0$, differentiation cancels the logarithmic denominator and gives $g_s=-(1/a)e^{ias}\Gamma(s+\sigma)t^{-s-\sigma}$. Integrating this <derivative> expresses $g(0)$ as a finite parameter-Laplace <integral> plus $g(s)$. The elementary lower bound $|ia+\log x|\ge a$ gives the displayed remainder, smaller than every logarithmic term at fixed $s>0$. <Watson's lemma> then determines all coefficients from the finite <integral>'s endpoint Taylor series.
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