The Fresnel integrals S(r) and C(r) can be defined as: S(t) ) = [Sir [' sin(x²)dr, C(t) = f' cos(x²) dr. We will use com

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The Fresnel Integrals S R And C R Can Be Defined As S T Sir Sin X Dr C T F Cos X Dr We Will Use Com 1 (67.86 KiB) Viewed 39 times

The Fresnel integrals S(r) and C(r) can be defined as: S(t) ) = [Sir [' sin(x²)dr, C(t) = f' cos(x²) dr. We will use complex integration to compute their limit at t → ∞o. (a) Consider f(z) = e. Find and classify all singularities of f(z), including at z = ∞0. [2] (b) Evaluate f ƒ(z)dz, where y is the wedge contour shown in Fig. 2 (0 ≤ 0 ≤ }). [2] (c) Using the previous result, show that -R R 1 * cos(2²) dx = -√2² e-²³ dr - Re ei=² dz R R J." [²* sin(x²)dx = 1/2 √² e-² dr - Im [8] (d) Prove that, on the circular are CR lim R+∞e e²dz = 0. √₁ 2 Hint: Jordan's inequality states that, if 0 ≤ x ≤, sinx ≥ = r. [10] ㅠ (e) Using the results of parts (c) and (d), show now that 1 lim S(t) lim C(t): = = [3] Hint: Remember · 5₁² € ²²² e dr V 2 Im z Re Los CR R Ice Rez et=² dz