Difference between revisions of "Fresnel S"
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(Created page with "The Fresnel $S$ function is defined by $$S(x)=\int_0^x \sin(t^2) dt.$$ <div align="center"> <gallery> File:Fresnel.png| Fresnel integrals on $\mathbb{R}$. </gallery> </div>") |
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File:Fresnel.png| Fresnel integrals on $\mathbb{R}$. | File:Fresnel.png| Fresnel integrals on $\mathbb{R}$. | ||
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| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following limit is known: | ||
| + | $$\displaystyle\lim_{x \rightarrow \infty} S(x) = \displaystyle\int_0^{\infty} \sin(t^2)dt = \sqrt{ \dfrac{\pi}{8}}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
</div> | </div> | ||
Revision as of 17:21, 10 March 2015
The Fresnel $S$ function is defined by $$S(x)=\int_0^x \sin(t^2) dt.$$
- Fresnel.png
Fresnel integrals on $\mathbb{R}$.
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: █
