# super cosx legendre series

I found this series could you give a explanation it converge very fast for $$-1<x<1$$ $$\cos (x)=\sum _{n=0}^{\infty } \frac{I_{n+\frac{1}{2}}(1) \left(2^{n-\frac{1}{2}} \left((-1)^n+1\right) e^{\frac{i \pi n}{2}} x^n \Gamma \left(n+\frac{3}{2}\right) \, _2F_1\left(\frac{1-n}{2},-\frac{n}{2};\frac{1}{2}-n;-\frac{1}{x^2}\right)\right)}{\Gamma (n+1)}$$ sorry for the english

• Found where?   – Kaster Mar 13 '15 at 19:09
• As with your other questions like this, it might be helpful to tell us where you got this from. – AlexR Mar 13 '15 at 19:18
• Are all the various special functions in the numerator really appreciably easier to compute than the cosine itself? Also at least some of it seems to be designed to obfuscate -- what's up with $((-1)^n+1)e^{i\pi n/2}$ (which works out to "zero for odd $n$, otherwise $2(-1)^{n/2}$), for example? – Henning Makholm Mar 13 '15 at 19:26