So i thought about applying the Watson lemma to determine the asymptotic behavior of the integral $$ I(x)=\int_{0}^{\infty} \frac{e^{-x(t-\ln(t))}}{(1+t^2)} dt, $$ as $x \rightarrow \infty$. I think it should be possible to substitute the term $t-\ln(t)$ so that we can apply the lemma. Has anyone an idea on how this could be worked out?
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2$\begingroup$ Hi and welcome to the Math Overflow. Premise: I am not the downvoter. However, I should point out that your question is better suited for our sister site Math.SE since this one is entirely devoted to research level mathematics Q&A, and your question is not research level. $\endgroup$– Daniele TampieriCommented Jan 28, 2023 at 14:25
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The first term in Watson's expansion amounts to the saddlepoint approximation (expansion of the exponent to second order around $t=1$), which gives $$I_{sp}=\frac{1}{2}e^{-x}\int_{-\infty}^\infty e^{-(x/2)(1-t)^2}\,dt=\frac{\sqrt{\pi } e^{-x}}{\sqrt{2 x}}.$$ This is quite accurate, see the plot (gold: exact; blue: saddlepoint approximation)
answered Jan 28, 2023 at 13:45
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$\begingroup$ Thank you. However, it would be nice if I could solve this with methods that I learned in class. :) $\endgroup$– helloCommented Jan 28, 2023 at 14:01