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Reciprocal-amplitude Laplace integral

Expanding

(1 + t)^{- 1}

at the endpoint and applying Watson's lemma gives

\int_{0}^{\infty} \frac{e ^{- x t}}{1 + t} d t \sim \sum_{n = 0}^{\infty} (- 1)^{n} n! x^{- n - 1} (x \to + \infty) .

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Solution