[The incomplete elliptic integral of the second kind](http://functions.wolfram.com/EllipticIntegrals/EllipticE2/introductions/IncompleteEllipticIntegrals/ShowAll.html) $E(\varphi \, | \,k)$ is defined as follows:


$$E(\varphi \, | \,k) = \int\_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $$


Where $0<k^2 < 1$.


Five years ago, [this](https://math.stackexchange.com/q/1123360/592365) MSE post was made asking about an inverse to this function (with respect to $\varphi$.) [Wolfram is (or, perhaps was, I am not sure) also looking for such a function](http://functions.wolfram.com/Contribute/topten.html#3).


I am posting this to ask if there has been any progress as to defining a function $E^{-1}$ so that
$$E(\varphi \, | \,k) = v \implies \varphi = E^{-1}(v \, | \,k)$$
Or approximating $E^{-1}(v \, | \,k)$.


**Edit:** [This](https://www.sciencedirect.com/science/article/abs/pii/S0096300311014809) article provides a numerial method to obtain a high-accuracy estimate of $E^{-1}(v \, | \,k)$.



Inverse of the incomplete elliptic integral of the second kind
For a numerical representation of the inverse in terms of the angle $\varphi$
where $E(\varphi \, | \,k) = \int\_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $ is the
elliptic integral for the second kind, one could expand $E(\varphi |k)$ in a power
series around $\varphi=0$,


$ E(k |\varphi) := \varphi
 -1/6 k^2\varphi^3
 +1/5(1/6k^2-1/8k^2)\varphi^5
 +1/7(-1/45 k^2+1/12k^4-1/16k^6)\varphi^7
 +1/9(1/630k^2-1/40k^4+1/16k^6-5/128k^8)\varphi^9 \cdots$


let $k^2=m$,The expansion coefficients in front of the order $\varphi^{n+1} ,(n=2,4,6,...)$ are
$\sum\_{r=2}^{n}U(r,n)F(r,m)/r!]/(n+1)!$, $r$ is even integer and 
$U(r,n) = (-1)^{(r+n)/2}/2^{r-n} \sum\_{l=0}^{r}(-1)^l (l-r/2)^n \binom{r}{l}]$ and $F(r,m) = -m^{n/2}[(r-1)!!]^2/(r-1)$ ,
with $(r-1)!! = 1\times 3\times 5\times 7\times \cdots\times(r-1) $ 


Then invert this as outlined in **chapt 3.6.25 of the book edited by M Abramowitz
and I Stegun** you get finally your inverse :


$\varphi := E(\varphi,m)
 +1/6mE(\varphi,m)^3
 +1/120m(13m-4)E(\varphi,m)^5
 +1/5040m(493m^2-284m+16)E(phi,m)^7
 +1/362880m(37369m^3-31224m^2+4944m-64)E(\varphi,m)^9 ...$


you may use Mathematica code up to $\varphi=12$ you may try the following code for numerical verification:



```
In[2]:= InverseSeries[Series[EllipticE[z, m], {z, 0, 12}]]
Out[2]= z + (m*z^3)/6 + (1/120)*(-4*m + 13*m^2)*z^5 + ((16*m - 284*m^2 +
493*m^3)*z^7)/5040 + ((-64*m + 4944*m^2 - 31224*m^3 +
37369*m^4)*z^9)/362880 + ((256*m - 81088*m^2 + 1406832*m^3 - 5165224*m^4 +
4732249*m^5)*z^11)/39916800 + O[z]^13

```

