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PostPosted: Thu Nov 02, 2017 9:35 pm 

Joined: Sat Nov 14, 2015 6:32 am
Posts: 132
Location: Melbourne, Australia
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.

Path
There are at least $2$ proofs. The one is more elegant than the other.

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$


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