# fractional iteration, Ramanujan

Ramanujan stated (quaterly reports, Ramanujan’s Notebooks vol. 1) that if $F(x) = x^{2} -2$, $x>2$ then the fractional iterates of $F$ are:

$F^{1/2}(x) = \left( \frac{x + \sqrt{x^{2}-4}}{2}\right)^{\sqrt{2}} - \left( \frac{x - \sqrt{x^{2}-4}}{2}\right)^{\sqrt{2}}$

$F^{\log(3)/\log(2)}(x) = x^{3} - 3x$

$F^{\log(5)/\log(2)}(x) = x^{5} - 5x^{3} +5x$

what’s kind of amazing about this is that some of $F$‘s transcendental order compositions are polynomials with rational coefficients!