Proving identity for Fresnel integrals

Prove the following identity for the Fresnel integrals:
lim ∫ sin(x²) dx = lim ∫ cos(x²) dx = sqrt(2π) /4
when R → ∞
where both integrals goes from 0 to R


Hint:
Integrate the function f(z) = e^(-z²) along the countour parametrized by

z(t) = Rt
when t ∈ [0, 1]

z(t) = Re^(2πit)
when t ∈ [1, 1 + ⅛]

z(t) = (2 + ⅛ - t)·Re^(iπ/4)
when t ∈ [1 + ⅛, 2 + ⅛]

and let R → ∞.



Thanks

Consider ∫c e^(-z²) dz, where C is the contour provided in the hint.

Since f(z) = e^(-z²) is entire, ∫c e^(-z²) dz = 0 by Cauchy's Theorem.
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On the other hand, breaking ∫c e^(-z²) dz into its three pieces:

Note: I have re-parameterized the curves a bit, it makes the computations toward infinity more transparent.
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(i) C₁: z(t) = t for t in [0, R]

∫c₁ e^(-z²) dz
= ∫(t = 0 to R) e^(-t²) dt

As R → ∞, this integral tends to ∫(t = 0 to ∞) e^(-t²) dt.

(ii) C₂: z(t) = Re^(it) for for t in [0, π/4]

∫c₂ e^(-z²) dz
= ∫(t = 0 to π/4) e^(-R² e^(2it)) * iRe^(it) dt

==> |∫c₂ e^(-z²) dz|
≤ ∫(t = 0 to π/4) |e^(-R² e^(2it)) * iRe^(it)| dt
= ∫(t = 0 to π/4) |e^[-R² (cos(2t) + i sin(2t))] * iRe^(it)| dt
= ∫(t = 0 to π/4) R e^[-R² cos(2t)] dt

As R → ∞, this integral tends to 0.
So, lim(R → ∞) ∫c₂ e^(-z²) dz = 0.

(iii) C₃: z(t) = te^(iπ/4) for t in [0, R] (clockwise orientation)

∫c₃ e^(-z²) dz
= -∫(t = 0 to R) e^(-(te^(iπ/4))²) * e^(iπ/4) dt
= -e^(iπ/4) ∫(t = 0 to R) e^(-t²e^(iπ/2)) dt
= -e^(iπ/4) ∫(t = 0 to R) e^(-it²) dt

As R → ∞, this integral tends to -e^(iπ/4) ∫(t = 0 to ∞) e^(-it²) dt
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Therefore as R → ∞,
0 = ∫c e^(-z²) dz
...= ∫(t = 0 to ∞) e^(-t²) dt + 0 - e^(iπ/4) ∫(t = 0 to ∞) e^(-it²) dt
...= (1/2)√π - e^(iπ/4) ∫(t = 0 to ∞) [cos(t²) - i sin(t²)] dt, via Gaussian Integral

Hence,
e^(iπ/4) ∫(t = 0 to ∞) [cos(t²) - i sin(t²)] dt = (1/2)√π
==> ∫(t = 0 to ∞) [cos(t²) - i sin(t²)] dt = (1/2)√π * e^(-iπ/4)
==> ∫(t = 0 to ∞) [cos(t²) - i sin(t²)] dt = (1/2)√π * (√2/2 - i√2/2)
==> ∫(t = 0 to ∞) [cos(t²) - i sin(t²)] dt = √(2π)/4 - i √(2π)/4.

Equating like entries yields
∫(t = 0 to ∞) cos(t²) dt = √(2π)/4, and ∫(t = 0 to ∞) sin(t²) dt = √(2π)/4.

I hope this helps!