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

(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

And

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

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

to complete the exercise.

Transcribed Image Text:

cos(x2 ) dx | sin(x2) dx = = 2V 2 cos(x2) dx = e-r'dr-Re - Reii wydx-ht-dr--ic.ee dz. sin(x2) CR r/2 -R2 sin¥d@ di JCR