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

$C(x):={\displaystyle {\int }_0^x}\cos t^{2}dt,S(x):={\displaystyle {\int }_0^x}\sin t^{2}dt.$ (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):

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

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

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

The parametric presentation

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

$${\int }_0^t\sqrt{C^{\prime }{(u)}^2+S^{\prime }{(u)}^2}𝑑u={\int }_0^t\sqrt{{\cos }^2(u^2)+{\sin }^2(u^2)}𝑑u={\int }_0^t𝑑u=t.$$

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