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In physics, some of the areas where you find elliptic integrals are:

In mathematics, elliptic integrals are part of the beautiful theory of elliptic functions. Elliptic integrals even show up in engineering when elliptic functions are used to design filters.

Since they are important in so many areas, alot of effort has gone into finding efficient ways of calculating elliptic integrals. In addition to calculating them, it is sometimes very useful to have a good approximation for them and the simpler the approximation, the better.

The simplest and most useful approximations are rational or Pade approximations. For a function of a single variable, such as the complete elliptic integrals, a rational approximation is just the ratio of two polynomials in the variable.

Below is a list of rational approximations for complete elliptic integrals of the first and second kind. These approximations are astonishingly good and they are the best rational approximations that we are aware of.

The derivation of these approximations has not yet been published. Once published, the derivation will be posted here. If you want to learn more about these approximations send Stefan an email (stefan at exstrom dot com).

Complete Elliptic Integrals of the First Kind

The following table lists rational approximations for complete elliptic integrals of the first kind. The larger the value of n, the better the approximation.

n q n(x)
3 x 2 −4 4 x 2 −8
5 3 x 4 −44 x 2 +64 18 x 4 −120 x 2 +128
7 x 8 −44 x 6 +432 x 4 −1280 x 2 +1024 8 x 8 −192 x 6 +1344 x 4 −3072 x 2 +2048
9 25 x 12 −2300 x 10 +50640 x 8 −403200 x 6 +1376256 x 4 −2031616 x 2 +1048576 250 x 12 −13000 x 10 +202400 x 8 −1267200 x 6 +3604480 x 4 −4587520 x 2 +2097152
11 9 x 18 −1656 x 16 +86032 x 14 −1933376 x 12 +21801984 x 10 −131715072 x 8 +438829056 x 6 −799014912 x 4 +738197504 x 2 −268435456 108 x 18 −12312 x 16 +470592 x 14 −8402816 x 12 +78594048 x 10 −406069248 x 8 +1184890880 x 6 −1925185536 x 4 +1610612736 x 2 −536870912

Below is a plot of the log of the difference between each of these approximations and the elliptic integral. The plot starts at x = 0.2 because below this value, the differences are so small that we had numerical instability problems trying to calculate them. Below x = 0.2, the approximations are essentially equivalent to the elliptic integral.

Elliptic integral approximations

Below is a plot of the elliptic integral and the n=7 approximation. You can see that there is essentially no difference until x becomes greater than about 0.9.

Elliptic integral and n=7 approximation

Complete Elliptic Integrals of the Second Kind

Coming soon.