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.