Difference between revisions of "Fresnel S"

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File:Fresnelsplot.png| Graph of $S$.
 
File:Fresnelsplot.png| Graph of $S$.
File:Domain coloring fresnel s.png | [[Domain coloring]] of [[analytic continuation]] of Fresnel $S$.
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File:Complexfresnelsplot.png | [[Domain coloring]] of [[analytic continuation]] of Fresnel $S$.
 
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Revision as of 22:41, 23 May 2016

The Fresnel $S$ function is defined by $$S(z)=\int_0^z \sin(t^2) dt.$$ (Note in Abramowitz&Stegun it is defined differently.)

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

Fresnel C

Videos

The Fresnel Integral S(x) - How to integrate sin(x^2)

<center>$\ast$-integral functions
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