Asymptotic evaluation of $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$

preprint2020arXivOpen access

We consider the integral $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$ as a function of the positive integer $n$ . We show that there exists an asymptotic series in $\frac{1}{n}$ and compute the first terms of this series together with an explicit error bound.

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Asymptotic evaluation of $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$

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Asymptotic evaluation of ∫0∞(sin⁡xx)n dx\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx∫0∞​(xsinx​)ndx

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Asymptotic evaluation of $\int_0^\infty\left(\frac{\sin x}{x}\right)^n\;dx$