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%%% Primary Title: Fresnel integrals
%%% Primary Category Code: 02.30.-f
%%% Filename: FresnelIntegrals.tex
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%%% Owner: pahio
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\begin{document}
\subsection{The functions $C$ and $S$}
For any real value of the argument $x$, the {\em Fresnel integrals} $C(x)$ and $S(x)$ are defined as the integrals
\begin{align}
C(x) \;:=\; \int_0^x\cos{t^2}\,dt, \qquad
S(x) \;:=\; \int_0^x\sin{t^2}\,dt.
\end{align}
In optics, both of them express the intensity of diffracted light behind an illuminated edge.
Using the \htmladdnormallink{Taylor series}{http://planetphysics.us/encyclopedia/TaylorFormula.html} expansions of cosine and sine, we get easily the expansions of the \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} (1):
$$C(z) \,=\, \frac{z}{1}\!-\!\frac{z^5}{5\!\cdot\!2!}\!
+\!\frac{z^9}{9\!\cdot\!4!}\!-\!\frac{z^{13}}{13\!\cdot\!6!}\!+\!-\ldots$$
$$S(z) \,=\, \frac{z^3}{3\cdot1!}\!-\!\frac{z^7}{7\!\cdot\!3!}\!
+\!\frac{z^{11}}{11\!\cdot\!5!}\!-\!\frac{z^{15}}{15\!\cdot\!7!}\!+\!-\ldots$$
These converge for all complex values $z$ and thus define entire transcendental functions.\\
The Fresnel integrals at infinity have the finite value
$$\lim_{x\to\infty}C(x) = \lim_{x\to\infty}S(x) = \frac{\sqrt{2\pi}}{4}.$$
\subsection{Clothoid}
The parametric presentation
\begin{align}
x \,=\, C(t), \quad y = S(t)
\end{align}
represents a curve called {\em 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 \htmladdnormallink{diagram}{http://www.wakkanet.fi/~pahio/A/A/clothoid.png}).
The arc length of the clothoid from the origin to the point \,$(C(t),\,S(t))$\, is simply
$$\int_0^t\sqrt{C'(u)^2+S'(u)^2}\,du = \int_0^t\sqrt{\cos^2(u^2)+\sin^2(u^2)}\,du = \int_0^tdu = t.$$
Thus the length of the whole curve (to the point\,
$(\frac{\sqrt{2\pi}}{4},\,\frac{\sqrt{2\pi}}{4})$) is infinite.
The curvature of the clothoid also is extremely simple,
$$\varkappa \,=\, 2t,$$
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 straight portion of way must be bent to a turn:\, the zero curvature of the line can be continuously raised to the wished curvature.
\end{document}
