# Fresnel integrals

## 0.1 The functions $C$ and $S$

For any real value of the argument^{} $x$, the Fresnel integrals^{} $C(x)$ and $S(x)$ are defined as the integrals^{}

$C(x):={\displaystyle {\int}_{0}^{x}}\mathrm{cos}{t}^{2}dt,S(x):={\displaystyle {\int}_{0}^{x}}\mathrm{sin}{t}^{2}dt.$ | (1) |

In optics, both of them express the .

Using the Taylor series^{} expansions of cosine and sine (http://planetmath.org/ComplexSineAndCosine), we get easily the expansions of the functions^{} (1):

$$C(z)=\frac{z}{1}-\frac{{z}^{5}}{5\cdot 2!}+\frac{{z}^{9}}{9\cdot 4!}-\frac{{z}^{13}}{13\cdot 6!}+-\mathrm{\dots}$$ |

$$S(z)=\frac{{z}^{3}}{3\cdot 1!}-\frac{{z}^{7}}{7\cdot 3!}+\frac{{z}^{11}}{11\cdot 5!}-\frac{{z}^{15}}{15\cdot 7!}+-\mathrm{\dots}$$ |

These converge for all complex values $z$ and thus define entire transcendental functions.

The Fresnel integrals at infinity have the finite value

$$\underset{x\to \mathrm{\infty}}{lim}C(x)=\underset{x\to \mathrm{\infty}}{lim}S(x)=\frac{\sqrt{2\pi}}{4}.$$ |

## 0.2 Clothoid

$x=C(t),y=S(t)$ | (2) |

a curve called clothoid. Since the equations (2) both define odd functions^{}, the clothoid has symmetry about the origin. The curve has the shape of a “$\sim $”
(see this http://www.wakkanet.fi/ pahio/A/A/clothoid.pngdiagram).

The arc length^{} of the clothoid from the origin to the point $(C(t),S(t))$ is simply

$${\int}_{0}^{t}\sqrt{{C}^{\prime}{(u)}^{2}+{S}^{\prime}{(u)}^{2}}\mathit{d}u={\int}_{0}^{t}\sqrt{{\mathrm{cos}}^{2}({u}^{2})+{\mathrm{sin}}^{2}({u}^{2})}\mathit{d}u={\int}_{0}^{t}\mathit{d}u=t.$$ |

Thus the of the whole curve (to the point
$(\frac{\sqrt{2\pi}}{4},\frac{\sqrt{2\pi}}{4})$) is infinite^{}.

The curvature (http://planetmath.org/CurvaturePlaneCurve) of the clothoid also is extremely ,

$$\varkappa =\mathrm{\hspace{0.17em}2}t,$$ |

i.e. proportional (http://planetmath.org/Variation) to the arc lenth; thus in the origin only the curvature is zero.

Conversely, if the curvature of a plane curve varies proportionally to the arc length, the curve is a clothoid.

This property of the curvature of clothoid is utilised in way and railway construction, since the form of the clothoid is very efficient when a portion of way must be bent to a turn: the zero curvature of the line can be continuously raised to the wished curvature.

Title | Fresnel integrals |
---|---|

Canonical name | FresnelIntegrals |

Date of creation | 2014-07-11 21:15:59 |

Last modified on | 2014-07-11 21:15:59 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 24 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 30B10 |

Classification | msc 26A42 |

Classification | msc 33B20 |

Related topic | SineIntegral |

Defines | clothoid |