Maclaurin series involving an elliptic integral

Question

I have been asked to find the Maclaurin series expansion of a term involving an elliptic integral, I would be grateful for any help as I am unsure as to how to even start this question. The term I have to expand is (4/pi)* K (sin(x)) , in which K(sin(x)) represents a complete elliptic integral of the first kind with modulus sin(x).

Thanks in advance =D

Answer 1

The straightforward but tedious route involves composing the two Maclaurin series

$$\begin{align*} \frac2{\pi}K(m)&=\sum_{k=0}^\infty \binom{2k}{k}^2\left(\frac{m}{16}\right)^k\\ \sin^2 \alpha&=-\sum_{k=1}^\infty \frac{2^{k-1} \alpha^k}{k!}\cos\frac{k\pi}{2} \end{align*}$$

or equivalently, using the Faà di Bruno formula (e.g. via the Bell polynomials) to generate the coefficients.