Elliptic integral Puns

I recently came up with the following approximation for the circumference of an ellipse:

OP's Formula for the Approximate Circumference of an Ellipse

Note that values of n around 2.8 give decent approximations.

Let me try to walk you through my thought process on this.

I started with the general form of the equation for the arc-length of a polar curve:

Arc-Length of a Polar Curve

All of the terms in this integrand are known for an ellipse, but this is an elliptic integral which does not have a solution that I can calculate! Now, the side-stepping mentioned in the title comes in. Take a look at a plot of this integrand on a rectangular coordinate system:

Integrand v Theta for Ellipse where a=5, b=3

Using one of Ramanujan's approximate formulas for the circumference of an ellipse (in this case, an ellipse with semi-major and semi-minor axes of 5 and 3, respectively), we get an approx. circumference of 25.527.

Consider a "replacement" rectangle with this same area, and a base of 2pi. The height of this rectangle, H, is therefore this approximate area divided by 2pi. In this case, H = 4.063.

Same Plot as Previous with y = 4.063 Shown

Introducing a new value, h = H - b = 1.063 (for this case) allows us to see how much greater than the semi-minor axis value our H is.

h% shows us that value of h as a percentage between the values of the semi-axes:

Equation for h%

There is a relation between this h% value and the eccentricity of any ellipse. As a reminder, the eccentricity is:

Equation of the Eccentricity of an Ellipse

And here is the plotted relationship between the eccentricity and h% (which is directly related to the circumference of the ellipse):

[h&a

I am working on an investigation that requires finding a general solution for the surface area of revolution of a bezier curve. I honestly thought that this topic would be widely documented online, but the only approaches I can find are advanced and use elliptic integrals. Given that we need to find the integral of an equation in the form Sqrt(a0x10+a1x9+a2x8+...+a10), is there any exact solution using simpler but "clever" integration methods to solve (not an approximation such as trapezoidal sum though)?

I need some help to solve the following integral and isolate E in function of n.

(n * pi) / sqrt(2) = integral sqrt( E - sin(pi * y) ) dy from 0 to 1

LaTex version:

$ \frac{n\pi}{\sqrt{2}} = \int_0^{1} \sqrt{E - \sin{(\pi y)}} \dd y $

I don't have much to add that's not already in the title. The complete elliptic integral of the second kind gives a quarter of the circumference of an ellipse; why are the other ones also "elliptic" integrals if they don't have anything to do with ellipses? If they do have something to do with them, what is it?

PRETEXT (so that you know why I’m doing this): I’ve been making a calculator with JavaScript. Recently, I started implementing elliptic integrals. I decided to use the Carlson symmetric forms to compute the elliptic F and E functions, and I should have the elliptic Π function working pretty soon, assuming the formulas still hold up.

QUESTION: Is there any really good formula for finding the complete elliptic integral of the third kind? More specifically, the F function can be found w/ the arithmetic-geometric mean, the E function can be found w/ a combination of the AGM and its derivative. Is there any equivalent for the complete elliptic Π function? Or is my best bet to just use the incomplete elliptic integral and set φ=π/2?

Thank you!

Are there any useful references or resources that intuitively show how Jacobi Elliptic functions [sn, cn, dn, etc] are geometrically interpreted from properties of ellipses? And how the Jacobi Elliptic functions and integrals can be shown to be generalizations of circular trig functions? Thanks!

Hello everyone. I'm currently trying to solve an electrostatics problem (form of the field in an arbitrary point in space caused by a charged half ring) and after some work I find myself with elliptic integrals. One of them looks like this:

[; \int_0^{\pi } \frac{dx}{{(k-cos(x+a))}^{\frac{3}{2}}} ;]

where a is smaller or equal to pi, and k is a positive real number.

I'm not familiar with elliptic integrals, all the information I have found online only would work if the angle in the cosine was x only (which would force me to have a = 0, and I'm trying to avoid that to find the most general expression I can get)

My question is, how can I go around this? I tried with a change of variables y = x+a, but that would change the limits:

[; \int_a^{\pi-a } \frac{dy}{{(k-cos(y))}^{\frac{3}{2}}} ;]

And that won't do the trick either, considering all reults online where for integrals with lower limit 0 and upper limit pi, forcing a = 0 again. Any help? Thanks in advance.

I'm trying to find/understand full proof of the fact that ∫√(1-kt²)/(1-t²)dt cannot be expressed as an elementary function. I've seen Galois theory before but never really done anything with fields of functions.

Brian Conrad's "Integration in elementary terms" is one of the best things I've read on the subject (along with its reference, Rosenlicht's "Integration in finite terms"), but there is one part on pg. 8 that doesn't make sense to me:

> it suffices to prove that there does not exist an identity of the form > > 1/√P(x) = ∑ c g'/g + h' > > [... which] is equivalent to to the equality of meromorphic 1-forms > > dx/y = ∑ c dg/g + dh > > on the compact Riemann surface C associated to the equation y² = P(x), and for deg(P) > 2 the left side is a nonzero holomorphic 1-form on C. But a nonzero holomorphic 1-form on a compact Riemann surface never admits an expression as a linear combination of logarithmic meromorphic differentials dg/g and exact meromorphic differentials dh.

This last statement seems like just a rephrasing of what needs to be proven---if I knew why this was true for 1-forms I could probably make the same argument for 1/√P(x), but I don't.

I was wanted to use in an argument that the motion of a simple pendulum cannot be expressed using only elementary functions, but I'm not really sure about that. This mathematically translates to the doubt if incomplete elliptic integrals of the first kind always admit an elementary solution or not. I'm guessing quite the opposite, that almost never an elementary solution exists, but I couldn't find any proof.

Is there a better, more intuitive, & remember-able, way to do the (calculus-level) theory of elliptic integrals than the long drawn out, easily forgettable, process of establishing the properties of elliptic integrals that you would find in, say, Goursat:

https://archive.org/stream/coursemathanalys01gourrich#page/226/mode/2up

or this exercise in Zorich:

http://i.stack.imgur.com/NG7uk.png

Since I have only few experience in elementary number theory I have no idea what I could do here. What would be the easiest way to approach such a problem?

Thanks in advance.

Apprantely this fantastical expression for magnetic field of an off axis current loop: http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5Cmu_%7B0%7DIR%7D%7B4%5Cpi%7D%5Cint_%7B0%7D%5E%7B2%5Cpi%7D%5Cfrac%7B%5Cleft%5Bz%5Ccos%5Cphi%5E%7B%5Cprime%7D%5Chat%7B%5Cmathbf%7Bx%7D%7D%2Bz%5Csin%5Cphi%5E%7B%5Cprime%7D%5Chat%7B%5Cmathbf%7By%7D%7D%2B%5Cleft%28R-y%5Csin%5Cphi%5E%7B%5Cprime%7D%5Cright%29%5Chat%7B%5Cmathbf%7Bz%7D%7D%5Cright%5Dd%5Cphi%5E%7B%5Cprime%7D%7D%7B%5Cleft%28R%5E%7B2%7D%2Br%5E%7B2%7D-2yR%5Csin%5Cphi%5E%7B%5Cprime%7D%5Cright%29%5E%7B3%2F2%7D%7D+&bg=eedbbd&fg=000000&s=0

from:

http://physicspages.com/2013/04/18/magnetic-field-of-current-loop-off-axis-field/ somehow reduces to this: http://www.netdenizen.com/emagnet/offaxis/iloopoffaxis.htm (the site is wrong, the k should not be square rooted)

mathematica game me this crazy indefinite integral, which is clearly more complicated than it needs to be https://i.imgur.com/i5MOew3.jpg

i am interested, how do i learn to write similar integrals as elliptic integrals?

The problem is following:

4. A parabolic satellite dish is 2m wide and 1/2m deep. Its axis of symmetry is tilted 30 degrees from the vertical. a. Set up, but dot not evaluate, a triple integral in rectangular coordinates that gives the amount of water the satellite dish will hold. (Hint: Put your coordinate sytem so that the satellite dish is in ”standard position” and the plane of the water level is slanted.)

The problem is also seen here. I don't have a subscription, so I only see step 1. My task is slightly different from the problem I stated since I need to evaluate the integral as well, which has proven troublesome for me.

Here are the questions I have about the problem:

1. It is said in the beginning that the unit vector normal to the plane has v=(0,a,b) where b is cos30(which I understand) and a is -sqrt(1-b^2). Is there any reason in particular the - sign in front of sqrt is necessary, or is it based on the arbitrary choice which way the satellite is slanted?

2. When I'm trying to evaluate the integral(using different online calculators), I keep getting complex answers, which leads me to believe I've made mistakes. In the final integral on the page provided earlier, the second integral(dy) is bounded by a function of y, which does not make sense to me. I figured the second integral is actually integrated with respect to x, but solving it this way leads to the complex number answer(which does not fit the problem).

Can anyone point me towards a direction what to try next? I've been stuck on this problem for a while and I can't seem to figure it out.

Like for example if you integrate something and get an elliptic integral inside like this, and you substitute a certain value of x into the integration result, will you get a number? for example for the photo I attached, if I said x = 5 and subbed that in will I get a number as the answer or what? I am not exactly sure what F(X|m) notation actually means

Hi /r/learnmath,

I'm trying to understand how Maxwell's 2nd formula for the mutual inductance of filaments can be transformed from the form seen in equation 11 to the form seen in equation 12.

I somewhat understand that the expression K(k) - E(k) = K(k)*C_s, and that K(k) can be approximated by AGM to be pi/2a_n. However, I don't fully get how K(k) is transformed to its final form, nor how these equations are manipulated.

The main goal of this is to use these concepts to calculate the inductance of multi-layered solenoids.

Is Mathematica capable of doing symbolic integration for elliptic coordinates? If so, do you have an examples of implementation or tutorials? If not, could you point me toward any software that can?

Thanks in advance!

Hi guys,

I'm having trouble setting up the polar integral after I integrate with respect to y (from 3-x to x-3). Normally if it was a circular cylinder the integral with respect to r would be from 0 to r, but in this case, it is an ellipse, so I don't know how to express the upper bound. I'm not sure if I should be using polar form in the first place, or another method? Thanks in advance.

https://imgur.com/1yxiHFe

https://preview.redd.it/4713j7aaq7551.png?width=2500&format=png&auto=webp&s=eaae6624d28326713acfa9249d6d991a0cebcb28

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Of all the shapes out there, I’ve noticed quite allot of effort goes into the study of ellipses and ellipse related objects. Why is this?

This is prompted by u/llcoolade03's post about math trivia that doesn't require calculations. I just got so excited by writing down a few of them I wanted to make a post of everything I have. Just a fair warning, I want to say these are fairly difficult, so proceed with caution.

1. What did Isaac Newton originally call calculus? >!Fluxions!<
2. What mathematician was the student of Georgio Ricci-Curbastro and famous for developing tensor analysis and assisting Einstein with the math used in his general theory of relativity? >!Tullio Levi-Civita!<
3. What language does the word algebra come from? >!Arabic!<
4. How many complex analysts have been convicted of murder? >!2, Ted Kazinsky and Andre Bloch!<
5. What the most widely accepted system of axioms of set theory?>! Zermelo-Frankael or ZF!<
6. What mathematician was famous for discovering an inconsistency in Naive set theory? >!Bertrand Russel!<
7. What logician was famous for proving that math will always be inconsistent or incomplete? >!Kurt Godel!<
8. What female mathematician was famous for her advancements in abstract algebra in the early 20th century?>! Emmy Noether!<
9. Who proved that there is no general solution to the quintic equation? >!Neils Abel!<
10. What theorem states that the distribution of random events affected by many unrelated factors has an approximately normal distribution? >!Central Limit theorem!<
11. What concept forms the foundation of calculus? >!Limits!<
12. What algorithm used to calculate the digits of pi is based on Ramanujan’s pi formula? >!Chudnovsky’s algorithm!<
13. What is Ramanujan’s constant? >!1729!<
14. What British mathematician proved Fermat’s Last theorem? >!Andrew Wiles!<
15. What theorem is a vast generalization of the fundamental theorem of calculus? >!Stokes Theorem!<
16. What Greek mathematician first proved that the square root of 2 is irrational using a proof by contradiction, something he is also famous for? >!Euclid!<
17. What French naturalist introduced the concept of a variable and developed the Cartesian plane? >!Rene Descartes!<
18. Who developed quaternions, an alternative to the matrices used in linear algebra? >!William Rowan Hamilton!<
19. Which famous American physicist/mathematician created a technique to evaluate integrals through interior differentiation, as well as make huge advancements in the many-body problem? >!Richard F

I have a surface fit and I want to evaluate the 2nd integral of this fit with an elliptically shaped base, i have my xmin and xmax but cant work out ymin(x) and ymax(x) integration limits for an ellipse? Thanks!

I've learned that their name comes from their usefulness in calculating the perimeter of an ellipse (I read that only a few integrals have this property, but the entire group is called as such for whichever reason). But I do not understand the idea behind this at all. I know the definition, the equations, but I don't understand how they're actually related to ellipticity.

I very much counted on finding some graphic explanation, but to no avail. Perhaps it doesn't help that I've only ever thought of integrals as a representation of area or volume and we're actually dealing with a perimeter here.

The graphs of their integrands are not even elliptical. I usually try to visualise the whole process when dealing with geometrical problems, but this is way beyond my comprehension.

Freaking elliptic integrals - how do they work?!

EDIT: It has now been solved! https://arxiv.org/abs/2108.02640

I thought I'd make a new one, with one of the simplest currently unresolved Diophantine equations, as an excuse to talk about how it can be an opportunity to communicate things about mathematics that are not generally known.

https://thehighergeometer.wordpress.com/2021/07/27/diophantine-fruit/

Links are provided to MathOverflow/Math.SE for source mathematics and definitions, and discussion of the surrounding issues.

And yes, I reference the famous one secretly involving rational points on an elliptic curve, where the solutions have 80 digits.