How to compute complete elliptic integral of the first kind in explicit form using elementary functions?

Question

How to compute complete elliptic integral of the first kind in explicit form using elementary functions? If it is not possible to compute complete elliptic integral of the first kind in explicit way, then is it possible to define the result? How would you define solution?

Answer 1

If it is not possible to compute complete elliptic integral of the first kind in explicit way, then is it possible to define the result? How would you define solution?

There is a general definition of the integral $\int_a^b f(x)\,dx$ which works for any continuous function $f:[a,b]\to\mathbb R$ (and also for some functions that are not continuous). The integrals such as $$\int_0^{\pi/2} \frac{1}{\sqrt{2\cos^2\theta+3\sin^2\theta}}\,d\theta$$ and $$\int_0^{\pi/2} \sqrt{2\cos^2 (e^\theta)+3\sin^2 \sqrt{\theta}}\,d\theta$$ are perfectly well-defined, regardless of our ability to write down their value in any explicit way. For many integrals there is no explicit formula for the result, which usually calls for some sort of approximation.

As J.M. said, the AGM method is an efficient way to compute $K(k)$. See, for example,

• Landen's transformation
• Arithmetic–geometric mean
• Evaluation of the complete elliptic integrals by the AGM method
• AGM (Wolfram)