$T=2\pi \sqrt{2as+u^{2}}$

$\left\{\begin{matrix}a=\frac{T^{2}-4\left(\pi u\right)^{2}}{8\pi ^{2}s}\text{, }&s\neq 0\text{ and }\left(T=0\text{ or }|\frac{arg(T^{2})}{2}-arg(T)|<\pi \right)\\a\in \mathrm{C}\text{, }&\left(T=0\text{ and }s=0\text{ and }u=0\right)\text{ or }\left(T=-2\pi u\text{ and }s=0\text{ and }|\frac{arg(u^{2})}{2}-arg(-u)|<\pi \text{ and }u\neq 0\right)\text{ or }\left(T=2\pi u\text{ and }s=0\text{ and }|\frac{arg(u^{2})}{2}-arg(u)|<\pi \text{ and }u\neq 0\right)\end{matrix}\right.$