Trigonometric integral / Complex Analysis

[irycio]

Homework Statement

Calculate the integral $$\int\limits_0^{\infty} \cos{x^2} dx$$

This is the exercise from complex analysis chapter, so I guess I should change it into a complex integral somehow and than integrate. I just don't know how, since neither substitution cos(x^2) = Re{e^ix^2} nor cos(x^2)=1/2(e^ix^2 + e^-(ix^2)) works.

Thanks in advance for your help :)

 

[CompuChip]

Do you know the Gaussian integral
$$\int_{-\infty}^\infty e^{-a x^2} \, dx = \sqrt{\frac{\pi}{a}}$$

It formally applies to all a for which Re(a) > 0, although as far as I am aware it continues to hold when Re(a) goes to 0, if Im(a) is non-zero, i.e.
$$\lim_{\epsilon \downarrow 0} \int_{-\infty}^\infty e^{-(\epsilon + i a) x^2} \, dx = \sqrt{\frac{\pi}{ ia}} = (1 - i) \sqrt{\frac{\pi}{a}}$$