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# arc length example

The functions

$x\;\mapsto\;\ln\sin{x}\quad\mbox{and}\quad x\;\mapsto\,\ln\cos{x}$ |

belong to the few real functions, the arc length of which are expressible in closed form (other ones are mentioned in the entry arc length of parabola).

We calculate the arc length of the curve

$y\;=\;\ln\sin{x}\qquad(0\;<\;a\;\leqq\;x\;\leqq\;\frac{\pi}{2}).$ |

By the chain rule, we have

$y^{{\prime}}\;=\;\frac{1}{\sin{x}}\cdot\cos{x}\;=\;\cot{x}.$ |

Hence the arc length is

$s\;=\;\int_{a}^{{\frac{\pi}{2}}}\!\sqrt{1+(\cot{x})^{2}}\,dx\;=\;\int_{a}^{{% \frac{\pi}{2}}}\!\frac{1}{\sin{x}}\,dx\;=\;\operatornamewithlimits{\Big/}_{{\!% \!\!a}}^{{\,\quad\frac{\pi}{2}}}\!\ln|\tan\frac{x}{2}|\;=\;\ln 1-\ln|\tan\frac% {a}{2}|\;=\;\ln\cot\frac{a}{2}$ |

(see integration of rational function of sine and cosine).

Related:

SubstitutionNotation

Synonym:

logarithm of sine function

Type of Math Object:

Example

Major Section:

Reference

Parent:

Groups audience:

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## Comments

## "ln" in pstricks

Hi, I would like to add a pstricks graph to

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

but I cannot implement for it the logarithm operator. Who could help?

Jussi

## Re: "ln" in pstricks

Can you say a little more about what you are trying to do and what is not working?

Roger

## Re: "ln" in pstricks

Dear Roger,

It seems that pstricks don't know the logarithm operators "ln" and "log". I have now made a simple test code in the entry -- you can see that logarithms don't work. Other operators, as "sin", "exp" etc., function well.

Jussi

## Re: "ln" in pstricks

Actually, if you look at the error messages closely, you will see that the problem is not with pstricks (PostScript does have a "ln" operator), but rather PM's automatic linking to the word "ln" is interfering. So you can just suppress the automatic linking and the diagram will work.

Steve

// Steve

## Re: "ln" in pstricks

Steve, you are fully right -- thank you very much!

I still wonder why the system doesn't allow to continue the logsin curve to the interval [Pi/2, Pi].

Regards,

Jussi

P. S. -- How about your German project?

## Re: "ln" in pstricks

> Steve, you are fully right -- thank you very much!

> I still wonder why the system doesn't allow to continue the

> logsin curve to the interval [Pi/2, Pi].

Sorry, no idea.

> P. S. -- How about your German project?

Unfortunately, I haven't got time. Incidentally the two other software developers in my group both speak German, and one is a Finn (PhD from Courant Institute); Jussi --- you are from Finland too right?

Something that might be interesting to PM'ers: Recently I've been working on optimized computations of transcendental functions, like exp, ln, normal cdf inverse, for use in Monte Carlo simulations (to value financial instruments). We found the implementations of the functions in system libraries were too slow, and because of our environment, the non-open-source replacements from CPU vendors were unsuitable. Knowing nothing about the subject, I had to dig up some research papers, and made notes. Hopefully, with permission, I will be able to publish them.

// Steve

## Re: "ln" in pstricks

Yes, I'm from Finland.