I am trying to write Mathematica code that evaluates the following function:
$$ f(\kappa_{yx}, \kappa_{zx}) = 1 + 3 \kappa_{yx} \kappa_{zx} \frac{E(\varphi \backslash \alpha) - F(\varphi \backslash \alpha)}{(1-\kappa_{zx}^2)\sqrt{1-\kappa_{yx}^2}} $$
where $E(\varphi \backslash \alpha)$ and $F(\varphi \backslash \alpha)$ are the incomplete elliptic integrals of the first and second kind, and $\varphi$ and $\alpha$ are defined according to:
$$ \sin{\varphi} = \sqrt{1-\kappa_{yx}^2} $$
$$ \sin^2{\alpha} = \frac{1-\kappa_{zx}^2}{1-\kappa_{yx}^2} $$
$\kappa_{yx}$ and $\kappa_{zx}$ are non-negative real numbers (i.e. can take on values greater than one). I need to evaluate the function $f$ because it appears in the equation of motion I'm trying to numerically integrate.
The paper I found the function $f$ in claims it is smooth and ranges from $-2$ to $1$, however Mathematica fails to evaluate the function at $\kappa_{yx} = 1$, $\kappa_{zx}=1$, or any values for which $\kappa_{yx}>1$.
I'm wondering if anyone can provide any guidance on how I can implement the function so that it returns real values for $\kappa_{yx}$ and $\kappa_{zx}$ greater than or equal to unity.
For reference, the paper provides the following plot of the function ( Log plot of function $f$ )
The paper also includes a polynomial approximation for the function that works well around unity but I need to evaluate the function when the arguments are around 2 or 3.
Here's the link to the paper: https://journals.aps.org/pra/pdf/10.1103/PhysRevA.74.013621