PlanetPhysics/Fresnel 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: .

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

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

$S(x)=\underset{0}{\overset{x}{\int }}\sin t^{2}dt.$

The functions C and S

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

Using the [[../TaylorFormula/|Taylor series]] expansions of cosine and sine, we get easily the expansions of the [[../Bijective/|functions]]:

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

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

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

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

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

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

Clothoid

The parametric presentation

$\begin{array}{c}x=C(t),y=S(t)\end{array}$

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 $\underset{0}{\overset{t}{\int }}\sqrt{C^{\prime }(u{)}^2+S^{\prime }(u{)}^2}du=\underset{0}{\overset{t}{\int }}\sqrt{{\cos }^2(u^2)+{\sin }^2(u^2)}du=\underset{0}{\overset{t}{\int }}du=t.$ 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.