PlanetPhysics/Fresnel integrals

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S(x) and C(x) The maximum of C(x) is about 0.977451424. If πt²/2 were used instead of t², then the image would be scaled vertically and horizontally (see below). Credit: .
Normalised Fresnel integrals, S(x) and C(x) have the argument of the trigonometric function is πt2/2, as opposed to just t2 as above. Credit: .

For any real value of the argument , the Fresnel integrals and are defined as the integrals:


The functions and [edit]

In optics, both of them express the intensity of diffracted light behind an illuminated edge.

Using the Taylor series expansions of cosine and sine, we get easily the expansions of the functions:

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

The Fresnel integrals at infinity have the finite value


The parametric presentation

represents a curve called clothoid .

Since the equations both define odd functions, the clothoid has symmetry about the origin.

The curve has the shape of a "" (see this diagram).

The arc length of the clothoid from the origin to the point ,, is simply Thus, the length of the whole curve to the point, is infinite.

The curvature of the clothoid also is extremely simple, 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 straight portion of way must be bent to a turn, the zero curvature of the line can be continuously raised to the wished curvature.