Difference between revisions of "Fresnel S"
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The Fresnel $S$ function is defined by | The Fresnel $S$ function is defined by | ||
$$S(z)=\int_0^z \sin(t^2) dt.$$ | $$S(z)=\int_0^z \sin(t^2) dt.$$ | ||
| + | (Note in Abramowitz&Stegun it [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_300.htm is defined] differently.) | ||
<div align="center"> | <div align="center"> | ||
Revision as of 10:30, 30 December 2015
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)
