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The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and are thus well-known special functions, and the same holds for some elliptic integrals. But what is the relation between the named elliptic functions and the named elliptic integrals?

It seems that the Jacobi amplitude $\varphi=\operatorname{am}(u,k)$ is the inverse of the elliptic integral of the first kind, $u=F(\varphi,k)$. Or related to this, $x=\operatorname{sn}(u,k)$ is the inverse of $u=F(x;k)$. It looks to me as if all of Jacobi's elliptic functions relate to the elliptic integral of the first kind. For other named elliptic functions listed by Wikipedia, like Jacobi's $\vartheta$ function or Weierstrass's $\wp$ function, it is even harder to see a relation to Legendre's integrals.

Is there a way to express the inverse of $E$, the elliptic integral of the second kind, in terms of some named elliptic functions? I.e. given $E(\varphi,k)=u$, can you write a closed form expression for $\varphi$ in terms of $k$ and $u$ using well-known special functions and elementary arithmetic operations?

In this post the author uses the Mathematica function FindRoot to do this kind of inversion, but while reading that post, I couldn't help wondering whether there is an easier formulation. Even though the computation behind the scenes might in fact boil down to root-finding in any case, it feels like this task should be common enough that someone has come up with a name for the core of this computation.

MvG
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  • This paper might be interesting for you: [*Numerical computation of incomplete elliptic integrals of a general form*](http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994CeMDA..59..237F&link_type=ARTICLE&db_key=AST&high==) by T. Fukushima and H. Ishizaki. Also the paper [*Numerical computation of inverse complete elliptic integrals of first and second kinds*](http://dx.doi.org/10.1016/j.cam.2013.02.003) by T Fukushima. – Loreno Heer Jan 28 '15 at 16:07
  • yes inverse calculations are in the second paper – Loreno Heer Jan 28 '15 at 16:08
  • @sanjab: Thanks for these pointers. But as far as I can see, the first paper only does forward computation, and the second does inverse (using notation like $m_E$), but of the *complete* integral and solving for the parameter $m$ not the amplitude $\varphi$. So at least as far as I can see, this is related but won't immediately help answering my question. – MvG Jan 28 '15 at 16:29
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    FWIW, [Wolfram is looking for something nice, too](http://functions.wolfram.com/Contribute/topten.html#3). – J. M. ain't a mathematician May 17 '16 at 03:40
  • N.B. the inverse of the Weierstrass elliptic function can of course be expressed in terms of $F$, or Carlson's $R_F$ if you prefer the symmetric integrals. Similarly, the Weierstrass and Jacobi functions are expressible in terms of each other (corresponding to the algebro-geometric notion of a quartic being equivalent to a cubic under suitable rational transformations.) – J. M. ain't a mathematician May 17 '16 at 03:43
  • There is a function called [InverseEllipticE](https://resources.wolframcloud.com/FunctionRepository/resources/InverseEllipticE/) in the function repository. – Tyma Gaidash Jan 10 '22 at 03:23
  • Would a differential equation for the function work? – Tyma Gaidash Feb 14 '22 at 01:47
  • @TymaGaidash At least for the use cases I have in mind, I don't see a differential equation adding value over the equation $E(\varphi,k)=u$ I used. In both cases you would not be able to use the formulation as a term in a larger expression. In both cases you could use it as a concise definition with a few words explaining what to do with it and how it connects to the rest of a computation. So they feel roughly equivalent in their utility from a notational perspective. – MvG Feb 15 '22 at 14:30

3 Answers3

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Here is a closed form for the inverse of functions containing $\text E(x,k),k=-1,2$ with it using the Mathematica function Inversebetaregularized $\text I^{-1}_z(a,b)$ which is a quantile function for the Student T Distribution with the Regularized $\text I_z(a,b)$ Beta function $\text B_z(a,b)$. Please see the links for efficiency. Here are the 3 main cases that are solvable, but inverses for other values of $k\ne 0,1$ probably have no closed form:

1.Notice that:

$$\frac14 \text B_{\sin^4(x)}\left(\frac34,\frac12\right)\mathop=^{0\le x\le \text L_2}\text E(x,-1)-\text F(x,-1)=\text D(x,-1)=z\implies x=\sin^{-1}\left(\sqrt[4]{\text I^{-1}_{\frac z{\text L_2}}\left(\frac34,\frac12\right)}\right)$$

which is correct. Also note the EllipticD function.

Graph of inverse:

enter image description here

2.Also note that:

$$\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)\mathop=^{0\le x\le \text L_2}\text E(x,2)=z\implies x=\frac12\sin^{-1}\left(\sqrt{\text I^{-1}_{\frac z{\text L_2}}\left(\frac12,\frac34\right)}\right)$$

which is correct

enter image description here

3.Similarly:

$$\frac58 \text B_{\sin^2(2x)}\left(\frac32,\frac34\right)=\text E(x,2)-\frac12\sin(2x)\cos^\frac32(2x)=z\implies x\mathop= ^{0\le x\le \text L_2}\frac12\sin^{-1}\left(\sqrt{\text I^{-1}_{\frac z{\text L_2}}\left(\frac32,\frac34\right)}\right)$$

enter image description here

which also works

There are also other special cases, but they are very specific and may not have applications. The $\text I^{-1}_z(a,b)$ function also gives special cases as Jacobi Elliptic functions with $k=-1,\frac12,2$. Also use the periodic nature of the elliptic integrals to find values outside of the $x$ restriction. Also, $\text L_2$ is The Second Lemniscate Constant Please correct me and give me feedback!

Tyma Gaidash
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I've just found a physical problem (in classical mechanics) involving the trajectory of a particle in which I had to take the inverse of $E(\phi,k)$, the incomplete elliptic integral of the 2nd kind. For the elliptic integral of the 1st kind $F(x,k)$ this is an easy task beacuse the Jacobi elliptic function $sn(x,k)$ (or JacobiSN(x,k) in mathematical software) is just $F^{-1}(x,k)$. However, currently there is no built-in function for the inverse of $E(\phi,k)$. My computational solution was then to build a procedure for the inverse, using FindRoot (in Mathematica9.0) or fsolve (in Maple 2015). F.M.S. Lima (University of Brasilia).

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I know this isn't a closed form, but I was interested in this question and have found one can relate the two functions together as series representations. I have written a short article here but this is the crux of it

Write \begin{equation} E(\phi,k) = \sum_{i=0}^\infty \frac{Q_i(k)}{(2i-1)!}\phi^{2i-1} \end{equation} where $Q_i(k)$ are polynomials in $k$, from the series expansion here we can get a finite form for these polynomials as \begin{equation} Q_n(m) = 2(-4)^n\sum_{k=1}^n \frac{(2k-3)!!}{k!}\left(\frac{-m}{8}\right)^k \sum_{j=0}^{k-1} \binom{2k}{j}(-1)^{1-j}(j-k)^{2n}, \;\;\; n>0 \end{equation} with $Q_0(k)$ defined as $1$. Then you can write the inverse series using series reversion in a very similar manner to $E(\phi,k)$ \begin{equation} \phi(E,k) = \sum_{i=0}^\infty \frac{R_i(k)}{(2i-1)!}E^{2i-1} \end{equation} where the relation between the new polynomials $R_i(k)$ is given by the explicit reversion formula found at the bottom of this link, giving \begin{equation} R_n(k) = (2n)! \sum_{\tau_n=n}(-1)^{\sigma_n} \frac{\prod_{i=1}^{\sigma_n}2n+i}{\prod_{j=1}^n k_j!}\prod_{l=1}^n \left(\frac{Q_l(k)}{(2l+1)!}\right)^{k_l} \end{equation} where $\sigma_n=k_1+k_2+k_3+\cdots+k_n$, and $\tau_n=k_1+2k_2+3k+3+\cdots + nk_n$ and the sum is over all sets of indices $k_i$ that meet the requirement $\tau_n=n$. I don't know if any nice simplifications or tricks can be made to reduce this to a functional form. The only numerical element here is converging the series to the desired accuracy.

Benedict W. J. Irwin
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  • The incomplete elliptic integral of the second kind is known to be expressible in terms of a (bivariate) hypergeometric function, so the appearance of factorials and binomial coefficients is not that surprising. The inverse obtained from Lagrangian inversion, however, does not seem to show any known pattern in the coefficients. – J. M. ain't a mathematician Jul 31 '17 at 06:10