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# Fresnel integrals

# 0.1 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

$\displaystyle C(x)\;:=\;\int_{0}^{x}\cos{t^{2}}\,dt,\quad\quad S(x)\;:=\;\int_% {0}^{x}\sin{t^{2}}\,dt.$ | (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):

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

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

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

The Fresnel integrals at infinity have the finite value

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

# 0.2 Clothoid

$\displaystyle x\;=\;C(t),\quad\quad y\;=\;S(t)$ | (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

$\int_{0}^{t}\sqrt{C^{{\prime}}(u)^{2}+S^{{\prime}}(u)^{2}}\,du=\int_{0}^{t}% \sqrt{\cos^{2}(u^{2})+\sin^{2}(u^{2})}\,du=\int_{0}^{t}du=t.$ |

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.

## Mathematics Subject Classification

30B10*no label found*26A42

*no label found*33B20

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## Comments

## uploading diagram file

Hi, it seems that the system for uploading files to the file box of an entry

(now http://planetmath.org/encyclopedia/FresnelIntegrals.html)

does not function. I would like add the clothoid picture

http://www.wakkanet.fi/~pahio/A/A/clothoid.png

to the entry. Can anyone help?

Jussi

## Re: uploading diagram file

I downloaded and then uploaded, so it's in your file list now.

Jon

## Re: uploading diagram file

Jon,

I am very grateful to you for uploading the picture. You master the thing since you have uploaded so many your own diagrams. Thanks!

Jussi

## Re: uploading diagram file

Jon, I got the error

"line 84: LaTeX Error: Cannot determine size of graphic in clo"

when I added to the code of

http://planetmath.org/encyclopedia/FresnelIntegrals.html

the lines

\begin{center}

\includegraphics[scale=0.5]{clothoid.png}

\end{center}

and tried to preview. Is it so that the system does not accept the png graphics files?

Jussi

## Re: uploading diagram file

I tried both SVG and JPEG once and it didn't work. I can't recall if I ever tried PNG, but I know I finally settled on EPS for some reason. Do you have the option of exporting to EPS? If not I'll see if the graphics package I have can convert from PNG to EPS somehow, but in the morning, as it's late here now.

Jon

## Re: uploading diagram file

> Do you have the option of exporting to EPS? If not I'll see if the graphics package I have can convert from PNG to EPS somehow, but in the morning, as it's late here now.

Jon, I know nothing about EPS (and PNG:).

Jussi

## Re: uploading diagram file

My old StarOffice Drawing package did a conversion from PNG to EPS, but it looks like there was some loss of quality.

I scaled it to the default 100% as the 50% reduction looked kind of small.

Jon

## Re: uploading diagram file

Jon,

Thank you very much for the clothoid diagram in EPS! (Was the 50% reduction sharper?)

Regards,

Jussi

## clothoid picture

Hi Jon, you have succeeded to get the clothoid and also your pictures in sight! Thanks! My picture is quite coarse though, but it has been such already before, too.

Jussi

## Re: clothoid picture

> Hi Jon, you have succeeded to get the clothoid and also your

> pictures in sight! Thanks! My picture is quite coarse

> though, but it has been such already before, too.

> Jussi

Who me? I didn't do anything but constant complaining.

It's my specialty.

I've been trying to convert the PNG to EPS again on a newer machine, but haven't succeeded in getting a better quality image yet.

Jon

## Re: clothoid picture

Dear Jon,

Now I saw that the same damned devil has broken another entry containing graphics (http://planetmath.org/encyclopedia/CurvatureOfNielsensSpiral.html)! It is evident that all such disasters are caused by the new server machine =o(

What should I do?

Jussi