Derive Fresnels integrals: 0 cos(x 2 ) dx = 0 sin(x 2 ) dx
Chapter 22, Problem 22.51(choose chapter or problem)
Derive Fresnels integrals: 0 cos(x 2 ) dx = 0 sin(x 2 ) dx = 1 2 2 . Hint: Integrate ei z2 over the closed path bounding the sector 0 x R, 0 /4, as in Figure 22.3. Use Cauchys theorem to evaluate this integral, then evaluate it as the sum of the integrals over the boundary segments of the sector. Show that the integral over the circular arc tends to zero as R , and use the integrals over the line segments to obtain Fresnels integrals. x y R /4
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