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FresnelC

The Fresnel Cosine Integral

FresnelS

The Fresnel Sine Integral

Fresnelf, Fresnelg

The Fresnel Auxiliary Functions

 Calling Sequence FresnelC(x) FresnelS(x) Fresnelg(x) Fresnelf(x)

Parameters

 x - algebraic expression

Description

 • The Fresnel cosine integral is defined as follows:

$\mathrm{FresnelC}\left(x\right)={\int }_{0}^{x}\mathrm{cos}\left(\frac{\mathrm{\pi }{t}^{2}}{2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$

 • The Fresnel sine integral is defined as follows:

$\mathrm{FresnelS}\left(x\right)={\int }_{0}^{x}\mathrm{sin}\left(\frac{\mathrm{\pi }{t}^{2}}{2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$

 • The Fresnel auxiliary functions are defined as follows:

$\mathrm{Fresnelf}\left(x\right)=\left(\frac{1}{2}-\mathrm{FresnelS}\left(x\right)\right)\mathrm{cos}\left(\frac{1}{2}\mathrm{\pi }{x}^{2}\right)-\left(\frac{1}{2}-\mathrm{FresnelC}\left(x\right)\right)\mathrm{sin}\left(\frac{1}{2}\mathrm{\pi }{x}^{2}\right)$

$\mathrm{Fresnelg}\left(x\right)=\left(\frac{1}{2}-\mathrm{FresnelC}\left(x\right)\right)\mathrm{cos}\left(\frac{1}{2}\mathrm{\pi }{x}^{2}\right)+\left(\frac{1}{2}-\mathrm{FresnelS}\left(x\right)\right)\mathrm{sin}\left(\frac{1}{2}\mathrm{\pi }{x}^{2}\right)$

Examples

 > $\mathrm{FresnelS}\left(\mathrm{\infty }\right)$
 $\frac{{1}}{{2}}$ (1)
 > $\mathrm{FresnelC}\left(1\right)$
 ${\mathrm{FresnelC}}{}\left({1}\right)$ (2)
 > $\mathrm{evalf}\left(\right)$
 ${0.7798934004}$ (3)
 > $\mathrm{Fresnelf}\left(1.0\right)$
 ${0.2798934004}$ (4)