I am trying to compute the real part of the following complex function:
$$S(z) = \frac{8}{3}\sqrt{z^{-\frac{1}{2}} + z^{-1}}\left(z + z^{\frac{1}{2}}\right)$$
For context, this expression was obtained from the following ordinary differential equation
$$y'' + \left(1 + 2z^{-\frac{1}{2}}\right)y' + \frac{1}{4}y = 0$$
for which I am trying to find the Stokes lines.
My first step is to obtain the controlling factor of the relevant leading behaviour. I assume, in the usual fashion, that $y(z) = e^{S(z)}$, from which I get the following equation in $S(z)$:
$$S''(z) + S'(z)^2 + \left(1 + 2z^{-\frac{1}{2}}\right)S'(z) + \frac{1}{4} = 0$$
Assuming that the $S''(z)$ term becomes negligibly small for $|z| \to \infty$, we get the relation
$$S'(z)^2 + \left(1 + 2z^{-\frac{1}{2}}\right)S'(z) + \frac{1}{4} = 0$$
This can be solved as a quadratic. I used Wolfram for the unpleasant integration. Taking the difference in the solutions, I get the above expression that I wish to take the real part of.
I am pretty much stuck though. Do I try and invoke Euler's formula? Perform some trick with the $z$'s?
Help or hints very much appreciated.