PlanetPhysics/Fresnel integrals



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


 * $$C(x) \;=\; \int_0^x\cos{t^2}\,dt,$$ 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:

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

$$S(z) \,=\, \frac{z^3}{3\cdot1!}\!-\!\frac{z^7}{7\!\cdot\!3!}\! +\!\frac{z^{11}}{11\!\cdot\!5!}\!-\!\frac{z^{15}}{15\!\cdot\!7!}\!+\!-\ldots$$


 * $$S(z)=\int_0^z \sin(t^2)\,\mathrm{d}t=\sum_{n=0}^{\infin}(-1)^n\frac{z^{4n+3}}{(2n+1)!(4n+3)}$$
 * $$C(z)=\int_0^z \cos(t^2)\,\mathrm{d}t=\sum_{n=0}^{\infin}(-1)^n\frac{z^{4n+1}}{(2n)!(4n+1)}$$

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 $$\int_0^t\sqrt{C'(u)^2+S'(u)^2}\,du = \int_0^t\sqrt{\cos^2(u^2)+\sin^2(u^2)}\,du = \int_0^tdu = t.$$ 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.