# Fresnel S in terms of erf

The following formula holds: $$S(z)=\sqrt{\dfrac{\pi}{2}} \dfrac{1+i}{4} \left[ \mathrm{erf} \left( \dfrac{1+i}{\sqrt{2}} z \right) - i \mathrm{erf} \left( \dfrac{1-i}{\sqrt{2}} z \right) \right],$$ where $S$ denotes Fresnel S, $\pi$ denotes pi, $i$ denotes the imaginary number, and $\mathrm{erf}$ denotes the error function.