How to find the first few terms in the asymptotic expansion of the given integral, as z tends to infinity, $$\int_0^\infty dt \left\{\frac{t-(e^ t -1)}{t(e^ t-1)}+1/2 \right\}e^{-tz} $$ Does this possess any relation to digamma function?
2026-04-03 11:04:08.1775214248
Asymptotic expansion of the given integral
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Near $t=0$ an expansion is $$\frac{t-(e^t-1)}{t(e^t-1)}+\frac{1}{2}\sim \frac{t}{12}-\frac{t^3}{720}+\frac{t^5}{30240}... $$ then, Watson lemma applies $$ I\sim \frac{1}{12z^2}-\frac{1}{120z^4}+...$$