Why isnt Cauchy's formula used for the perimeter of ellipse?

[Austin Daniel]

So the formula for an ellipse in polar coordinates is r(θ) = p/(1+εcos(θ)). By evaluating L = ∫r(θ) dθ on the complex plane on a circle of circumference ε on the centered at the origin I obtained the equation L = (2π)/√(1-ε^2). Why then does Wikipedia say that the formula for the perimeter is an elliptic integral? I thought we couldn't come up with a closed form solution to an elliptic integral? I also saw someplace on the internet that this is only an approximate solution? Is Cauchy's formula only an approximation?

 

[mathman]

Your formula for L is not the length of the perimeter. Correct formula is $L=\int_{0}^{2\pi }\sqrt{r^2+(\frac{dr}{d\theta})^2}d\theta$.

 

[SteamKing]

So the formula for an ellipse in polar coordinates is r(θ) = p/(1+εcos(θ)). By evaluating L = ∫r(θ) dθ on the complex plane on a circle of circumference ε on the centered at the origin I obtained the equation L = (2π)/√(1-ε^2). Why then does Wikipedia say that the formula for the perimeter is an elliptic integral? I thought we couldn't come up with a closed form solution to an elliptic integral? I also saw someplace on the internet that this is only an approximate solution? Is Cauchy's formula only an approximation?

A closed form solution to an integral means that it can't be expressed in terms of simpler functions, like polynomials, logarithms, or trig functions. The definite integral can be evaluated, however, by various numerical techniques and other approximations. Mathematical handbooks contain tables of the values of the various elliptic integrals and elliptic functions.

 

[mathwonk]

the division line between elementary and non elementary is entirely arbitrary. Even if an integral could be evaluated using cosines and sines or even square roots, any numerical evaluation would still be approximate. e.g. the square root of 5 is only approximable numerically. I.e. there is nothing inherently more elementary about the power series for cosine than the series for an elliptic function to me at least, maybe I am naive. Of course elliptic functions tend to have poles, maybe that's the problem?

But in my opinion we are sort of brainwashed into thinking we understand trig functions and logarithms just because we hear about them early. I.e. when we write the answer to our problem as a trig function we breathe a little sigh of relief and apt ourselves on the back. They are still quite difficult to deal with precisely. It is not always pointed out to young students that they can't know the value of sin or cos anywhere except at very simple angles. We also pretend in calculus that we know fundamental things like how to multiply two real numbers together, when actually we don't even have a nice notation for even writing down a typical real number; i.e. it's at best an infinite non repeating decimal. But don't mind me, it took me a long time to realize many of us have been teaching calculus of real valued functions for years without even explaining to students what a real number is.

Of course trig and logarithm functions do satisfy some nice addition and multiplication formulas, but so do elliptic functions - these were discovered by Abel. in geometry we learn elliptic functions are functions deined on elliptic curves, or curves of genus one. so in the great panoply of curves, of arbitrary positive genus, elliptic curves are pretty elementary and vastly simpler than a typical curve, say of genus 6, or genus 7,892.

So in the larger realm of function theory we may start from the definition of the exponential function as the inverse of the integral of 1/x, and the trig functions as inverses of integrals of 1/sqrt(1-x^2), or of 1/sqrt(quadratic(x)); an elliptic function then is the inverse of the integral of 1/sqrt(cubic or quartic(x));

We thus see that elliptic functions are not very far out on the limb of possibilities for this construction. More generally any function obtained by inverting an integral of form 1/sqrt(polynomial(x)) is called a "hyprelliptic function". You can see now that even these are pretty special since only a square root is involved.

In fact hyperelliptic functions, like say the inverse of the integral of 1/sqrt(1-x^500), are so special that I almost always excluded them by hypothesis (perhaps foolishly) from my research investigations into general curves. I.e. after genus 2, most curves of genus g are not even hyperelliptic. Still they are sufficiently elementary that they serve as a good source of examples that can be worked with more concretely, as Mumford showed when he worked out the structure of the jacobian of any hyperelliptic curve, in a more specific way than before. This model has since been generalized to more general curves.

My point is just that ellliptic and even hyperelliptic curves, and integrals, are elementary from a certain point of view, at least in a relative sense, compared to what is out there in general.

 

[SteamKing]

the division line between elementary and non elementary is entirely arbitrary. Even if an integral could be evaluated using cosines and sines or even square roots, any numerical evaluation would still be approximate. e.g. the square root of 5 is only approximable numerically. I.e. there is nothing inherently more elementary about the power series for cosine than the series for an elliptic function to me at least, maybe I am naive. Of course elliptic functions tend to have poles, maybe that's the problem? But in my opinion we are sort of brainwashed into thinking we understand trig functions and logarithms just because we hear about them early. They are still quite difficult to deal with precisely. It is not always pointed out to young students that they can't know the value of sin or cos anywhere except at very simple angles. We also pretend in calculus that we know fundamental things like how to multiply two real numbers together, when actually we don't even have a nice notation for even writing down a typical real number; i.e. it's at best an infinite non repeating decimal. But don't mind me, it took me a long time to realize many of us have been teaching calculus of real valued functions for years without even explaining to students what a real number is.

Of course trig and logarithm functions do satisfy some nice addition and multiplication formulas, but so do elliptic functions - these were discovered by abel. in geometry we learn elliptic functions are functions deined on elliptic curves, or curves of genus one. so in the great panoply of curves, of arbitrary positive genus, elliptic curves are pretty elementary and vastly simpler than a typical curve, say of genus 6, or genus 7,892.

If you don't like math, just say so already!

 

[mathwonk]

well i am writing some notes on "solving" differential equations, and i am suddenly wondering what it means to claim to have "solved" an equation just because one can say the answer is e^πt. That function can only be evaluated by an infinite series, hence only approximately, and because of the π in there, not even the second term is precise in that series. I just think we should be honest about what we are saying to students. In "solving" problems or equations we often just give a name to a solution we know almost nothing about. It is always prudent to ask just what we know about our answer. Can we evaluate it numerically? do we know some nice abstract properties? Or have we just related two different things about both of which we know little?

e.g. i always learned the solution of f' = af is c.e^at, so that equation looks easy to me, when a and c are constants. But what about the system x' = A.x, where x is a vector valued function and A is a matrix? Nobody told me about just plugging the matrix At into the same infinite series for e^t nd getting a matrix called e^At, so when told the answer to the system is just e^(At).c, where c is a vector, I shrank back in fear, because e^matrix was not as elementary to me at least as e^number. But you approximate both of them in reality, one is just more complicated. And in fact if A is diagonal, e^At is the same as e^at, just repeated down the diagonal.

so beware of terms like "elementary", ask for the definition and what is meant. i am wondering if here we don't just mean "simpler", or "something I learned in school". But there is no firm line between elementary and non elementary as is often pretended. i.e. what is elementary to holmes may not be to watson.

closed form also means nothing. why is e^t considered a closed form of 1 + t + t^2/2 +... you could give a finite name to any infinite expression, like 1 + 1/2^3 + 1/3^3 +1/4^3 +... = "alice". (or zeta(3)).

 

[SteamKing]

I think this article gives a handle on what "closed form" means for certain mathematical purposes:

https://en.wikipedia.org/wiki/Closed-form_expression

Similarly, elementary functions:

https://en.wikipedia.org/wiki/Elementary_function

 

[mathwonk]

here is the trick phrase in that article:

"The set of operations and functions admitted in a closed-form expression may vary with author and context."

so you are allowed to pretend that cos(1) is evaluated in finite form.

 

[SteamKing]

here is the trick phrase in that article:

"The set of operations and functions admitted in a closed-form expression may vary with author and context."

so you are allowed to pretend that cos(1) is evaluated in finite form.

The values of the trig functions may be irrational in some cases, if not transcendental, but that doesn't take away from the fact that a definite value exists.

If our number system is incapable of expressing this value in a finite number of digits or whatever, that's a separate matter altogether. To me, it's like saying we don't (or can't) know what π is because we can't write out all the digits.

No one has claimed that the so-called elementary functions we study (like exponentials, logarithms, and the like) give nice round results when evaluated, because they don't. Infinite series give an alternate representation of these functions which can extend the precision to an arbitrary degree by evaluating additional terms. This is also certainly true of functions (like elliptic integrals) which are evaluated by approximating the value of a definite integral through whatever convenient numerical method is at hand.

 

[mathwonk]

precisely. the point is that the so called elementary functions are on exactly the same footing as the esoteric or non elementary ones. both require approximation.