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Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?

How to calculate $\displaystyle\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$ ?

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marked as duplicate by DonAntonio, Sasha, Pedro Tamaroff, Davide Giraudo, Micah Sep 21 '12 at 20:50

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

have a look at this Wolfram Alpha Solution – Ayush Khemka Sep 21 '12 at 18:31

If you have some familiarity with complex analysis, this hint may be helpful:

Recall the Gaussian integral, defined $\forall \alpha \in \mathbb{R}: \int_0^\infty e^{-\alpha x^2} dx = \sqrt{\frac{\pi}{4\alpha}}$.

Now consider $e^{ix^2} = \cos(x^2) + i \sin(x^2)$. Naively, what does the above result suggest about the value of $\int_0^\infty e^{ix^2} dx$? If we take real and imaginary parts of the integral and its value, what would we find?

Most importantly, what contour integral in $\mathbb{C}$ would justify this? (think geometrically about what branch and value was chosen to "compute" $\sqrt{\frac{1}{-i}}$.)

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