I have a function of the form
$$f(x;\lambda) = {}_2F_1\left(a,b;c;-\frac{e^{2x}}{\lambda}\right)$$
I need to invert this function to solve for the constant $\lambda = f\left(x\right)$. I could do this numerically using the Newton-Raphson or bisection methods but this is not efficient for me. There are so many hypergeometric transformations that I don't know where to start.
Any suggestions?