Follow the steps below to evaluate the Fresnel integrals, which

Follow the steps below to evaluate the Fresnel integrals, which are important in diffraction theory:
Follow the steps below to evaluate the Fresnel integrals, which

(a) By integrating the function exp(iz2) around the positively oriented boundary of the sector 0 ‰¤ r ‰¤ R, 0 ‰¤ θ ‰¤ Ï€/4 (Fig. 99) and appealing to the Cauchy-Goursat theorem, show that

Follow the steps below to evaluate the Fresnel integrals, which
Follow the steps below to evaluate the Fresnel integrals, which

And

Follow the steps below to evaluate the Fresnel integrals, which

Where CR is the arc z = Reiθ (0 ‰¤ θ ‰¤ Ï€/4).
(b) Show that the value of the integral along the arc CR in part (a) tends to zero as R tends to infinity by obtaining the inequality

Follow the steps below to evaluate the Fresnel integrals, which

and then referring to the form (2), Sec. 81, of Jordan's inequality.
(c) Use the results in parts (a) and (b), together with the known integration formula

Follow the steps below to evaluate the Fresnel integrals, which

to complete the exercise.