I am currently reading Heisenberg's PhD thesis. There is a common technique in fluid dynamics which keeps cropping up and I can never get my head around.
The equation of interest is the Orr-Sommerfeld equation
$(\phi '' - \alpha^{2}\phi)(w-c) - \phi w'' = \frac{i}{\alpha R}(\phi'''' - 2\alpha^2 \phi '' + \alpha^4 \phi )$
for which we seek an asymptotic representation of the form
$\phi = e^{\int g dy}, \quad g = \sqrt{\alpha R}g_{0} + g_{1} + \frac{1}{\sqrt{\alpha R}}g_{2} + ...$
The questions is;
Why is it in this particular form and not some other powers, say,
$\phi = e^{\int g dy}, \quad g = g_{0} + \frac{1}{\alpha R}g_{1} + \frac{1}{(\alpha R)^2}g_{2} + ...$
??