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## On the evaluation of the Fresnel integral

Real Analysis
Riemann
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### On the evaluation of the Fresnel integral

We are aware of the Fresnel integral

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

The most common proof goes with complex analysis. Try to provide a proof with Real Analysis.
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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$