The evaluation of the Fresnel integrals has been done a plethora of times both on this site, and numerous other places. The two main ways of evalutating these integrals has either been with some algebraic manipulations. $$ \int_{-\infty}^{\infty} \sin(x^2)\mathrm{d}x = \int_{0}^{\infty} \frac{\sin t}{\sqrt{t}}\mathrm{d}t =\frac{2}{\sqrt{\pi}}\int_0^\infty \int_0^\infty e^{-tx^2}\sin{t} \, \mathrm{d}x\,\mathrm{d}t =\frac{2}{\sqrt{\pi}}\int_0^{\infty} \frac{\mathrm{d}x}{1+x^4} =\sqrt{\frac{\pi}{8}} $$ Where it was amongst other things used that $1/\sqrt{t} = \pi^{-1}\int_{-\infty}^{\infty}e^{-tx^2}\mathrm{d}x$ and $\int_0^{\infty}e^{-\alpha t}\sin \beta t \mathrm{d}t = \beta/(\alpha^2 + \beta^2)$. Or for an example http://www.jstor.org/stable/2320230 , http://www.math.binghamton.edu/loya/papers/LoyaMathMag.pdf

Another path to take is complex analysis usually using path in the figure below


I have seen several proofs using the indented path, but the sources are few. For a proof se p.32 here or for a proof on stack.exchange see here. My question is asking for refferences for these methods to evaluate the Fresnel integrals and in particular using complex analysis.

What are the earliest occurrences of using complex analysis to evaluate the Fresnel integrals?


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