Fresnel S
From specialfunctionswiki
The Fresnel $S$ function is defined by $$S(z)=\int_0^z \sin(t^2) dt.$$ (Note in Abramowitz&Stegun it is defined differently.)
- Fresnel.png
Fresnel integrals on $\mathbb{R}$.
Domain coloring of analytic continuation of Fresnel $S$.
Properties
Theorem: The following limit is known: $$\displaystyle\lim_{x \rightarrow \infty} S(x) = \displaystyle\int_0^{\infty} \sin(t^2)dt = \sqrt{ \dfrac{\pi}{8}}.$$
Proof: █
See Also
Videos
The Fresnel Integral S(x) - How to integrate sin(x^2)
