I wish to approximate the following ode with Padé:
$$\dot \rho +i\frac{{\hat \sigma }}{2}\rho + i\alpha \rho \left[ { - {\gamma ^2} + \frac{{{\gamma ^3}}}{{\sqrt {{\gamma ^2} + 1} }}} \right] + i\frac{{\hat P}}{4}{\rho ^*} = 0$$
where $$\rho(t)$$ is a complex function of time and $$[~\gamma=\frac{1}{{\sqrt \beta \left| \rho \right|}}$$ $$\beta,$$$$\alpha,$$$$\hat \sigma,$$$$\hat P~]$$ are constants.
I developed Taylor's series about zero but I get a very inconvenient expression. How one can transform the ode into a convenient from before expanding to Taylor's?