This older answer of mine may contain material of interest

Also, about how large is the argument of $E(x)$ for your parameters (closer to 0 or 1?) It would also help if you stated which convention you're using for $E(x)$.

$x$ can range from 0 to 1 so how would I know how large my argument of $E(x)$ is? $4 E(x)=\int_0^{2 \pi } \sqrt{1-\frac{4 \pi ^2 \psi ^2 \left(\sin ^2(\theta)\right)}{\gamma ^2+4 \pi ^2 \psi ^2}} \, d\theta =4 \int_0^{\frac{\pi }{2}} \sqrt{1-x^2 \sin ^2(x)} \, d\theta$

Well, presumably you have a sense of how big $\gamma$ is compared to $2\pi \psi$ in your problem of interest.I stress that because you'll want to use different approximations if $x\approx 0$ vs $x \approx 1$.

Sorry I misjudged the situation, $x$ ergo $E(x)$ must be closer to 1 since when $\psi$ and $\gamma$ are close together, for example when $\psi=1$ and $\gamma=2$ $x \approx 0.952890514$ and when $\psi$ and $\gamma$ are further apart, for example when $\psi=10000$ and $\gamma=2$ $x\approx 0.9999999995$

So, to be clear, you want $x$ in the nbhd of one? The infinite series expression in $x^2$ is then particularly ill-suited; an infinite series in $1-x^2$, on the other hand, is far more appropriate. (Another way to put it is that you really want to deal with the complementary elliptic integral $E'(x)=E(\sqrt{1-x^2})$.) It would also be helpful if you could put some of that info into your question specifically so that your audience is on the same page.

My point was $x$ is always in the nbhd of 1, regardless of how close or far apart $\psi$ and $\gamma$ are. I want to be able to do what you did for your older answer for this problem.

Not if $\gamma \gg \psi$. In that case $x\approx 0$ (and in which case your series converges a good deal more quickly.)

Ok so I have two cases, where psi>>gamma -> x is in the nbhd of 0 and the aforementioned

Right. You'll want to approach them separately because they behave a bit differently. Let me grab a WolframAlpha plot quickly

wolfr.am/1uGMxir

That looks nicer than I expected, actually...

How would I start approximating this into its closed form? Can I do what you have done for your older answer?

You can, but it might be overkill. Need to mull over this a bit more.

Ok. First, something important for the case where psi is much bigger than gamma: a good first approximation is just that it's asymptotically linearly with $\psi/\gamma$. That's why it has that nice asymptote. (Compare wolfr.am/1s48xPV)

The next order correction is of the form log(x)/x, which disappears quickly enough

If you include that correction, though, you get a really nice approximation in the case of psi being nearly as large as gamma (compare wolfr.am/1uGOwTW)

(note that the x in that plot is 2*pi*psi with gamma=1 for convenience

very nice, I am just trying to follow you atm

sure, I know I'm going pretty quick

the gist is that there's probably no need to do any of the stuff from my previous answer here. (for the gamma >> psi case, perhaps)

Ok, but how much bigger?

hmm.

depends on how many digits you want, but let me find a representative plot

wolfr.am/1rG6S1U

There's three lines there: the elliptic integral is in yellow, the expansion to quadratic order near 0 in red, and the expansion to leading-log correction near infinity in blue

you'll note that the two approximations together rather well together (though that may be overstating things a bit, since the number of digits of precision isn't obvious)

So the blue can used for when psi>>gamma and psi\approx gamma (which seems to be the most accurate)? Also how do I increase the precision for these approximations so they aren't rough estimates?

one obvious way is to use more terms so that their range of accuracy improves (keeping in mind that you'll need some criterion for which approximation to use). Wolfram can give you that

the other approach is to do something like in my last answer re: AGM techniques. those are nice because they're iterative but they require more thought to implement

for the latter, this AMS article is a good reference point

So the curves are used for when psi>>gamma or psi is close to gamma, but what about gamma>>psi?

gamma>>psi is the small x behavior on the graph.

remembering that x=2pi*psi/gamma

i should probably have plotted wrt to t or something to avoid confusion, sorry

Oh ok so the red is used for that whereas the blue is used for the other case

right

for a numerical analysis, here's a nice plot showing the absolute differences between the two approximations and the exact answer

wolfr.am/1qKZoP0

(log plot of the absolute differences)

they do pretty well for most of the x values, and very well outside the interval [1,2]

one can improve these approximation with more terms; there may be a smarter way, but it'll require more thinking on my part

How did you come up with the 2 approximations?

"Sqrt[1+x^2]EllipticE[x^2/(1+x^2)] near 0" and "Sqrt[1+x^2]EllipticE[x^2/(1+x^2)] near infinity" in WolframAlpha. note it supplies additional terms upon request

these can presumably be found as well in reference work on the elliptic integrals

Thank you for your help

glad to help. I'll put together an answer compiling this info eventually (maybe with some AGM stuff as well, maybe not)