In this review on the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,

$$ r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta $$ where $g(\omega)$ is some unknown function (though is later taken to be the p.d.f. of the Lorentz distribution $g(\omega) = (\gamma/\pi)/(\gamma^2 + \omega^2)$, for some parameter $\gamma$). The asymptotic expression they obtain via the expansion is $$ r \sim \sqrt{\frac{8(K-K_{c})}{-K_{c}^{3}g''(0)}} $$ where $K_c = 2/(\pi g(0))$. I don't see how you can expand this integral in powers of $Kr$ (I think this mean a Taylor expansion in a new variable $z=Kr$ at $z=0$) without knowledge of $g$. Not much information is available, lots of other reviews appear to also omit this step. Is it a straightforward expansion? How is Eq. 14 obtained?

In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,

$$ r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta $$ where $g(\omega)$ is some unknown function (though is later taken to be the p.d.f. of the Lorentz distribution $g(\omega) = (\gamma/\pi)/(\gamma^2 + \omega^2)$, for some parameter $\gamma$). The asymptotic expression they obtain via the expansion is $$ r \sim \sqrt{\frac{8(K-K_{c})}{-K_{c}^{3}g''(0)}} $$ where $K_c = 2/(\pi g(0))$. I don't see how you can expand this integral in powers of $Kr$ (I think this means a Taylor expansion in a new variable $z=Kr$ at $z=0$) without knowledge of $g$. Not much information is available, lots of other reviews appear to also omit this step. Is it a straightforward expansion? How is Eq. 14 obtained?