I have seen the Fresnel integral
evaluated by contour integration and other complex analysis methods, and I have found these methods to be the standard way to evaluate this integral. I was wondering, however, does anyone know a real analysis method to evaluate this integral?
Aside from some trigonometric substitutions and identities, we will need the following identity, which can be shown using integration by parts twice:
We will also use the standard arctangent integral:
(3.1) change the square of the integral into a double integral
(3.2) use 2sin(A)sin(B)=cos(A−B)−cos(A+B)
(3.3) convert to polar coordinates
(3.4) substitute s=r2
(3.5) apply (1)
(3.6) pull out the constant and apply symmetry to reduce the domain of integration
(3.7) multiply numerator and denominator by sec2(2ϕ)
(3.8) substitute t=λtan(2ϕ)
(3.9) apply (2)
Finally, take the square root of both sides of (3) and let λ→0+ to get
I just noticed that the same proof works for
if each red sin is changed to cos and each red − sign is changed to +.