Table of Contents
Closed-form approximation of complete elliptic integrals
| Stan Zurek, Closed-form approximation of complete elliptic integrals, Encyclopedia Magnetica, https://www.e-magnetica.pl/doku.php/approximation_of_complete_elliptic_integrals |
Closed-form approximation of complete elliptic integrals
Elliptic integrals are useful in solving various mathematical problems, such as length of arc of an ellipse1), or calculating the force between two cylindrical magnets. However, the solution of en elliptic integral requires solving an integral (in analytical or numerical way) which makes their direct application much more difficult - because analytical precise closed-form equations do not exist.
However, it is possible to approximate the elliptic integral functions with other non-linear functions of the closed form, with some degree of accuracy. This page includes an example of such approximation with around 0.5% accuracy over most of the range of so-called complete elliptical function (upper limit set to π/2). The behaviour of the functions is illustrated in the figure, and the equations are included below.
S. Zurek, E-Magnetica.pl, CC-BY-4.0
Elliptic integral of first kind
| Elliptic integral of the first kind, K |
| $$ K(k) = \int_0^{π/2} \frac{1}{\sqrt{1 - k^2 · sin^2(θ)}} ~ dθ $$ |
| Closed-form non-linear approximation of K | |
| $$ K_{appr}(k) = \frac{π}{2 · (1-k)^{0.19}} - 0.17 · (k + 0.015)^{0.8} $$ | valid for $0 \leq k \leq 0.99999$ $ ε < 0.5$%, for $0 \leq k \leq 0.94$ $ ε < 10$%, for $k \leq 0.994$ |
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Elliptic integrals of second kind
| Elliptic integral of the first kind, E |
| $$ E(k) = \int_0^{π/2} \sqrt{1 - k^2 · sin^2(θ)} ~ dθ $$ |
| Closed-form non-linear approximation of E | |
| $$ E_{appr}(k) = \frac{π}{2} - 0.567 · k^{2.4 + (k+0.1)^{5.8}} $$ | valid for $0 \leq k \leq 1$ $ ε < 0.4$%, for $0 \leq k \leq 1$ |