Fresnel integrals

0.1 The functions C and S

For any real value of the argumentMathworldPlanetmathPlanetmath x, the Fresnel integralsDlmfDlmfDlmfDlmfMathworldPlanetmath C(x) and S(x) are defined as the integralsDlmfPlanetmath

C(x):=0xcost2dt,S(x):=0xsint2dt. (1)

In optics, both of them express the .

Using the Taylor seriesMathworldPlanetmath expansions of cosine and sine (, we get easily the expansions of the functionsMathworldPlanetmath (1):


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

The Fresnel integrals at infinity have the finite value


0.2 Clothoid

The parametric presentation

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

a curve called clothoid.  Since the equations (2) both define odd functionsMathworldPlanetmath, the clothoid has symmetry about the origin.  The curve has the shape of a “” (see this pahio/A/A/clothoid.pngdiagram).

The arc lengthMathworldPlanetmath of the clothoid from the origin to the point  (C(t),S(t))  is simply


Thus the of the whole curve (to the point  (2π4,2π4)) is infiniteMathworldPlanetmath.

The curvature ( of the clothoid also is extremely ,

ϰ= 2t,

i.e. proportional ( 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