How Do I Integrate e^{iz^2} to Obtain the Fresnel Integrals?

[e(ho0n3]

Homework Statement
Integrate [itex]e^{iz^2}[/itex] around the contour C to obtain the Fresnel integrals:

[tex]\int_0^\infty \cos(x^2) \, dx = \int_0^\infty \sin(x^2) \, dx = \frac{\sqrt{2\pi}}{4}[/tex]

The contour consists of three parts:

1. z = x, [itex]0 \le x \le R[/itex]
2. z = [itex]Re^{i\theta}[/itex], [itex]0 \le \theta \le \pi/4[/itex]
3. z = [itex]te^{i\pi/4}[/itex], [itex] R \ge t \ge 0[/itex]

The attempt at a solution
I'm stumped because I don't know how to evaluate integrals of the form [itex]e^{iz^2}[/itex]. How would I do this?

 

[Nino MMP]

My attempt:\int_C e^{iz^2} \, dz = \int_0^R e^{ix^2} \, dx + \int_0^{\pi/4} Re^{iRe^{i\theta}^2} \, iRe^{i\theta} \, d\theta + \int_R^0 te^{i\pi/4}e^{it^2e^{i\pi/2}} \, ite^{i\pi/4} \, dt I'm not sure how to evaluate the integrals in the middle and last terms.