# Fresnel integrals

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

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

 $\displaystyle C(x)\;:=\;\int_{0}^{x}\cos{t^{2}}\,dt,\quad\quad S(x)\;:=\;\int_% {0}^{x}\sin{t^{2}}\,dt.$ (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):

 $C(z)\,=\,\frac{z}{1}\!-\!\frac{z^{5}}{5\!\cdot\!2!}\!+\!\frac{z^{9}}{9\!\cdot% \!4!}\!-\!\frac{z^{13}}{13\!\cdot\!6!}\!+\!-\ldots$
 $S(z)\,=\,\frac{z^{3}}{3\cdot 1!}\!-\!\frac{z^{7}}{7\!\cdot\!3!}\!+\!\frac{z^{1% 1}}{11\!\cdot\!5!}\!-\!\frac{z^{15}}{15\!\cdot\!7!}\!+\!-\ldots$

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

The Fresnel integrals at infinity have the finite value

 $\lim_{x\to\infty}C(x)\;=\;\lim_{x\to\infty}S(x)\;=\;\frac{\sqrt{2\pi}}{4}.$

## 0.2 Clothoid

 $\displaystyle x\;=\;C(t),\quad\quad y\;=\;S(t)$ (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

 $\int_{0}^{t}\sqrt{C^{\prime}(u)^{2}+S^{\prime}(u)^{2}}\,du=\int_{0}^{t}\sqrt{% \cos^{2}(u^{2})+\sin^{2}(u^{2})}\,du=\int_{0}^{t}du=t.$

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 ,

 $\varkappa\,=\,2t,$

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 FresnelIntegrals 2014-07-11 21:15:59 2014-07-11 21:15:59 pahio (2872) pahio (2872) 24 pahio (2872) Definition msc 30B10 msc 26A42 msc 33B20 SineIntegral clothoid