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Fresnel integrals (Topic)

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

$\displaystyle C(x) \;:=\; \int_0^x\cos{t^2}\,dt, \qquad S(x) \;:=\; \int_0^x\sin{t^2}\,dt.$ (1)

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 (1):

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

$\displaystyle S(z) \,=\, \frac{z^3}{3\cdot1!}\!-\!\frac{z^7}{7\!\cdot\!3!}\! +\!\frac{z^{11}}{11\!\cdot\!5!}\!-\!\frac{z^{15}}{15\!\cdot\!7!}\!+\!-\ldots$
These converge for all complex values $z$ and thus define entire transcendental functions.

The Fresnel integrals at infinity have the finite value

$\displaystyle \lim_{x\to\infty}C(x) = \lim_{x\to\infty}S(x) = \frac{\sqrt{2\pi}}{4}.$

Clothoid

The parametric presentation

$\displaystyle x \,=\, C(t), \quad y = S(t)$ (2)
represents 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 diagram).

The arc length of the clothoid from the origin to the point   $(C(t),\,S(t))$  is simply

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

$\displaystyle \varkappa \,=\, 2t,$
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.



"Fresnel integrals" is owned by pahio.

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Keywords:  Fresnel integral, clothoid

Attachments:
Fresnel formulae (Theorem) by pahio

Cross-references: functions, Taylor series

This is version 1 of Fresnel integrals, born on 2009-04-18.
Object id is 648, canonical name is FresnelIntegrals.
Accessed 263 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
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