File:Fresnel Integral Contour.svg

The sector contour used to evaluate the limits of the Fresnel integrals. As R tends to infinity, this contour integral of the function ${\displaystyle e^{-{\frac {1}{2}}t^{2}}}$.

As R tends to infinity γ₂ tends to 0, and γ₁ tends to:

${\displaystyle \int _{0}^{\infty }e^{-{\frac {1}{2}}t^{2}}dt={\frac {\sqrt {2\pi }}{2}}={\sqrt {\frac {\pi }{2}}},}$