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On the evaluation of the Fresnel integral

Real Analysis
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Riemann
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On the evaluation of the Fresnel integral

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Post by Riemann » Thu Nov 02, 2017 9:35 pm

We are aware of the Fresnel integral

\begin{equation} \int_0^\infty \sin x^2 \, {\rm d}x = \frac{1}{2} \sqrt{\frac{\pi}{2}} \end{equation}

The most common proof goes with complex analysis. Try to provide a proof with Real Analysis.
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There are at least $2$ proofs. The one is more elegant than the other.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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