Difference between revisions of "Fresnel C"

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The Fresnel C function is defined by the formula
 
The Fresnel C function is defined by the formula
$$C(x)=\int_0^x \cos(t^2) dt.$$
+
$$C(z)=\int_0^z \cos\left(t^2\right) \mathrm{d}t.$$
 
+
(Note in Abramowitz&Stegun it [http://specialfunctionswiki.org/mirror/abramowitz_and_stegun-1.03/page_300.htm is defined] differently.)
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Fresnel.png| Fresnel integrals on $\mathbb{R}$.
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File:Fresnelcplot.png| Graph of $C$.
 +
File:Complexfresnelcplot.png|[[Domain coloring]] of Fresnel $C$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Fresnel C is odd]]<br />
<strong>Theorem:</strong> The following limit is known:
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[[Taylor series for Fresnel C]]<br />
$$\displaystyle\lim_{x \rightarrow \infty} C(x) = \displaystyle\int_0^{\infty} \cos(t^2)dt = \sqrt{ \dfrac{\pi}{8}}.$$
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[[Fresnel C in terms of erf]]<br />
<div class="mw-collapsible-content">
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[[Limiting value of Fresnel C]]<br />
<strong>Proof:</strong> █
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</div>
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=See Also=
</div>
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[[Fresnel S]]
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=Videos=
 +
[https://www.youtube.com/watch?v=fR4yd6pB5co How to integrate cos(x^2) - The Fresnel Integral C(x) (2 December 2014)]<br />
 +
[https://www.youtube.com/watch?v=H3uOq7VujYA Math and Physics: The Fresnel Integrals (12 May 2016)] <br />
  
 
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[[Category:SpecialFunction]]

Latest revision as of 05:10, 21 December 2017

The Fresnel C function is defined by the formula $$C(z)=\int_0^z \cos\left(t^2\right) \mathrm{d}t.$$ (Note in Abramowitz&Stegun it is defined differently.)

Properties

Fresnel C is odd
Taylor series for Fresnel C
Fresnel C in terms of erf
Limiting value of Fresnel C

See Also

Fresnel S

Videos

How to integrate cos(x^2) - The Fresnel Integral C(x) (2 December 2014)
Math and Physics: The Fresnel Integrals (12 May 2016)

$\ast$-integral functions