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Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of $\int_{-\infty}^{\infty}\exp(ix^2) $ but have no idea how to continue.

Knowing that $\int_{-\infty}^{\infty}\exp(-x^2) = \sqrt \pi$ one can feel that this value has to be somehow "divided" over the entire complex plane, so that every quadrant gets some part of $\sqrt \pi$. This would give the desired result of $\sqrt \frac{\pi}{2} - i \sqrt \frac{\pi}{2} $

Is there a way this could be proved or justified?

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Instead of "knowing" $\int_{-\infty}^{\infty}\exp(-x^2)\,dx=\sqrt{\pi}$ if you knew how this formula was derived, you'd be able to modify the argument to get what you want. So, look for a proof of the integral you know. –  Gerry Myerson Dec 5 '12 at 0:03

3 Answers 3

up vote 2 down vote accepted

Check out this answer for a real method to evaluate this integral.

First note that $$ \int_0^\infty e^{-x^2}\,\mathrm{d}x=\frac{\sqrt\pi}2\tag{1} $$

The integral $$ \int_\gamma e^{-z^2}\,\mathrm{d}z=0\tag{2} $$ over the curve $\gamma$ consisting of the line from $0$ to $R$ then counterclockwise along the circular arc from $R$ to $Re^{i\pi/4}$ then back along the line from $Re^{i\pi/4}$ to $0$ must be $0$ since $e^{-z^2}$ has no singularities inside $\gamma$.

Next note that $\int_\gamma e^{-z^2}\,\mathrm{d}z$ along the line from $0$ to $R$ as $R\to\infty$ tends to $(1)$.

Next note that $\int_\gamma e^{-z^2}\,\mathrm{d}z$ along the arc of the circle of radius $R$ from $R$ to $Re^{i\pi/4}$ as $R\to\infty$ goes to zero.

Finally, note that $\int_\gamma e^{-z^2}\,\mathrm{d}z$ along the line from $Re^{i\pi/4}$ to $0$ is $$ -e^{i\pi/4}\int_0^\infty e^{-ix^2}\,\mathrm{d}x\tag{3} $$ Therefore, $(1)$ plus $(3)$ is $0$, so we get $$ \int_0^\infty e^{-ix^2}\,\mathrm{d}x=\frac{1-i}{\sqrt2}\frac{\sqrt\pi}2\tag{4} $$ Taking the imaginary part of $(4)$ yields that $$ \int_0^\infty\sin(x^2)\,\mathrm{d}x=\sqrt{\frac\pi8}\tag{5} $$ or $$ \int_{-\infty}^\infty\sin(x^2)\,\mathrm{d}x=\sqrt{\frac\pi2}\tag{6} $$

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Starting from $\int_{-\infty}^\infty \exp(-x^2)\ dx = \sqrt{\pi}$, use a change of variables to get $\int_{-\infty}^\infty \exp(-ax^2)\ dx = \sqrt{\pi/a}$ for $a > 0$. Now both sides are analytic in $a$ for $\text{Re}(a) > 0$ (using a branch of the square root that is analytic in the right half plane), so the equation should still be true for $\text{Re}(a) > 0$. The next part is tricky to justify: you want to take the limit as $a \to -i$. Assuming this is valid, you have

$$ \int_{-\infty}^\infty \exp(ix^2)\ dx = \sqrt{\frac{\pi}{-i}}= (1+i) \sqrt{\frac{\pi}{2}}$$

so that $\int_{-\infty}^\infty \cos(x^2)\ dx = \int_{-\infty}^\infty \sin(x^2)\ dx = \sqrt{\pi/2}$.

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Hint: you should integrate $\exp(-x^2)$ along the Fresnel contour; the integral along the real axis you know, the integral along the diagonal you want to find, the arc at infinity vanishes.

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