What is Elliptic integral: Definition and 16 Discussions

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals. Originally, they arose in connection with the problem of finding the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form











{\displaystyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)\,\mathrm {d} t,}
where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.
In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when P has repeated roots, or when R(x,y) contains no odd powers of y. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).
Besides the Legendre form given below, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.

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  1. J

    I The asymptotic behaviour of Elliptic integral near k=1

    I'm looking at a proof of the asymptotic expression for the Elliptic function of the first kind https://math.stackexchange.com/questions/4064023/on-the-asymptotic-behavior-of-elliptic-integral-near-k-1 and I'm having trouble understanding this step in the proof: $$ \begin{align*} \frac{1}{2}...
  2. Robin04

    Solve Elliptic Integral: Tips & Ideas

    I need to solve this integral which I suppose is an elliptic integral but don't know what kind, I'm not that familiar with them. Mathematica says that it can be expressed with elementary functions and gives the solution: ## -\frac{2\...
  3. Javier Lopez

    Elliptic integral of the third order for magnet calculus

    I have the first and second orders that I use in a magnetic simulator, but i need the thirth also to do also with magnetic cylinders accordingly paper: Do anybody have it in any code? I should pass to C++
  4. StudentOfScience

    I Series for Elliptic Integral of the First Kind

    I'm not sure if this should go in the homework forum or not, but here we go. Hello all, I've been trying to find a series representation for the elliptic integral of the first kind. From some "research", the power series for the complete form (## \varphi=\frac{\pi}{2} ## or ## x=1 ##) seems to...
  5. S

    Spherical Pendulum - elliptic integral

    Hello everyone. In the 3rd edition of Mechanics by Landau and Lifshitz, paragraph 14, there is a problem concerning spherical pendulum. Calculations leading to the integral $$ t=\int \frac {d \Theta} {\sqrt{\frac{2}{ml^2}[E-U_{ef}(\Theta)]}},$$ $$...
  6. A

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

    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...
  7. X

    Inversion of incomplete elliptic integral of the second kind

    Hello I hope this is the right place to ask this question. For my thesis I need a way to invert a incomplete elliptic integral of the second kind. I believe the Jacobi elliptic functions are inverse of the elliptic integral of the first kind. The calculation I'm doing is symbolic so a...
  8. B

    MHB Definite integral on elliptic integral where modulus is function of variable

    How to prove: $\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $ where \[ k(\theta)=\sqrt\frac{4Rh \tan \theta}{Z^2+(R+h \tan \theta)^2}\] and $ K[k(\theta)] $ is the complete elliptic integral of the first kind...
  9. C

    Showing Complete Elliptic Integral of First Kind Maps to Rectangle

    Homework Statement Effectively, I'm trying to show the following two integrals are equivalent: \int_1^{1/k}[(x^2-1)(1-k^2x^2)]^{-1/2} dx = \int_0^1[(1-x^2)(1-(k')^2x^2)]^{-1/2}dx where k'^2 = 1-k^2 and 0 < k,k' < 1. Homework Equations One aspect of the problem I showed the following...
  10. S

    Elliptic integral and pseudo-elliptic integral from Wikipedia.

    Hi all. I was reading this Wikipedia article: http://en.wikipedia.org/wiki/Risch_algorithm I have a couple of questions about \int \frac{x}{\sqrt{x^4 + 10x^2 - 96x - 71}} \, dx and \int \frac{x}{\sqrt{x^4 + 10x^2 - 96x - 72}} \, dx discussed in the article. How the heck did they get the...
  11. G01

    Elliptic Integral Homework: Expanding for Large k^2

    Homework Statement Sub problem from a much larger HW problem: From previous steps we arrive at a complete elliptic integral of the second kind: E(k)=\int_0^{\pi/2} dx \sqrt{1+k^2\sin^2x} In the next part of the problem, I need to expand this integral and approximate it by truncating at the...
  12. M

    Change of Variables for Elliptic Integral

    Homework Statement Given the differential equation u_{xx}+3u_{yy}-2u_{x}+24u_{y}+5u=0 use the substitution of dependent variable u=ve^{ \alpha x + \beta y} and a scaling change of variables y'= \gamma y to reduce the differential equation to v_{xx}+v_{yy}+cv=0Homework Equations I have no...
  13. K

    Solving Elliptic Integral: Replacing x with 1/kx

    When replacing x with 1/kx then \int_{1/k}^\infty {\left[ {\left( {x^2 - 1} \right)\left( {k^2 x^2 - 1} \right)} \right]} ^{ - 1/2} dx = \int\limits_0^1 {\left[ {\left( {\frac{1}{{k^2 x^2 }} - 1} \right)\left( {\frac{1}{{x^2 }} - 1} \right)} \right]} ^{ - 1/2} \frac{{dx}}{{kx^2 }} I...
  14. P

    Elliptic Integral Homework: Calculate \int_{0}^{\pi/2}\frac{dx}{\sqrt{\sin{x}}}

    Homework Statement The problem is to calculate integral \int_{0}^{\pi/2}\frac{dx}{\sqrt{\sin{x}}} by transforming it into elliptical form (complete elliptical integral of first kind).
  15. S

    Mathematica showing strange output for elliptic integral

    Hi, I was studying calculus and I had a problem while checking my results. I came to the following result: \int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin t}}\:\mathrm{d}t = \sqrt{2}\cdot \mathrm{F}\left(\frac{\pi}{4},\frac{1}{4}\right) \approx1.16817 However, Mathematica shows...
  16. A

    Elliptic Integrals: Arc Length of Ellipses and Elliptic Curves

    Taken from http://en.wikipedia.org/wiki/Elliptic_integral: Is it just me, or does it seem like there is an easier way to find the arc length of an ellipse? I thought elliptic integrals arose in giving the arc length of elliptic curves, which as far as I know are a lot different than ellipses.