PlanetPhysics/Fresnel integrals

For any real value of the argument $x$, the Fresnel integrals $C(x)$ and $S(x)$ are defined as the integrals:

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: .

$C(x)\;=\;\int _{0}^{x}\cos {t^{2}}\,dt,$ and

$S(x)\;=\;\int _{0}^{x}\sin {t^{2}}\,dt.$

The functions C and S

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:

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

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

$S(z)=\int _{0}^{z}\sin(t^{2})\,\mathrm {d} t=\sum _{n=0}^{\infty }(-1)^{n}{\frac {z^{4n+3}}{(2n+1)!(4n+3)}}$

$C(z)=\int _{0}^{z}\cos(t^{2})\,\mathrm {d} t=\sum _{n=0}^{\infty }(-1)^{n}{\frac {z^{4n+1}}{(2n)!(4n+1)}}$

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

The Fresnel integrals at infinity have the finite value $\lim _{x\to \infty }C(x)=\lim _{x\to \infty }S(x)={\frac {\sqrt {2\pi }}{4}}.$

Clothoid

The parametric presentation

${\begin{matrix}x\,=\,C(t),\quad y=S(t)\end{matrix}}$

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 "$\sim$" (see this diagram).

The arc length of the clothoid from the origin to the point ,$(C(t),\,S(t))$, is simply $\int _{0}^{t}{\sqrt {C'(u)^{2}+S'(u)^{2}}}\,du=\int _{0}^{t}{\sqrt {\cos ^{2}(u^{2})+\sin ^{2}(u^{2})}}\,du=\int _{0}^{t}du=t.$ 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.