Migration and whatnot in past 10 months

Posted on November 27, 2020

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$\begin{aligned} r_{1}=&\frac{(k\lambda{}\ell+d^{2})^{2}+4d^{2}\ell^{2}}{4d^{2}}\\ =&\frac{k^{2}\lambda{}^{2}\ell^{2}+2d^{2}k\lambda{}\ell+d^{4}+\ell^{2}}{4d^{2}}\\ =&\frac{k^{2}\lambda{}^{2}\ell^{2}+2d^{2}(k\lambda{}\ell{}+d^{2}+2\ell^{2})}{4d^{2}}\\ =&\frac{k^{2}\lambda{}^{2}\ell^{2}+2d^{2}[2\ell{}(k\lambda{}+\ell)+d^{2}]}{4d^{2}}\\ \approx{}&\frac{k^{2}\lambda{}^{2}\ell^{2}+2d^{2}(2\ell{}^{2}+d^{2})}{4d^{2}}\\ \approx{}&\frac{k^{2}\lambda{}^{2}\ell^{2}+2d^{2}(2\ell{}^{2})}{4d^{2}}\\ ={}&\frac{k^{2}\lambda{}^{2}\ell^{2}+4d^{2}\ell{}^{2}}{4d^{2}} \end{aligned}$