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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=Properties=
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<strong>Theorem:</strong> The following limit is known:
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$$\displaystyle\lim_{x \rightarrow \infty} S(x) = \displaystyle\int_0^{\infty} \sin(t^2)dt = \sqrt{ \dfrac{\pi}{8}}.$$
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<strong>Proof:</strong> █
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Revision as of 17:21, 10 March 2015

The Fresnel $S$ function is defined by $$S(x)=\int_0^x \sin(t^2) dt.$$

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: