deletions | additions       diff --git a/main.tex b/main.tex  index c3ae212..0bb6af6 100644  --- a/main.tex  +++ b/main.tex 
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and plotting convergent values with changing $r$. Looking back this is now obvious, as when solving the integral and changing $\alpha$ to $r$ we have the iterative form \begin{equation}  [I_r(\beta)]_{i+1}= r[I_r(\beta)]_{i}(1 - \frac{\beta}{2r} [I_r(\beta)]_{i}) \end{equation}  From this we can say fit the following functions,  \begin{equation} I_r(\beta)=\int_0^{I_r(\beta)} r - \beta x \; \mathrm{dx} = \frac{2}{\beta}(r-1), \;\; 1\le x \le 3 \\  I_r(\beta)=\int_0^{I_r(\beta)} r - \beta x \; \mathrm{dx} = \Bigg\{  \begin{matrix}  4(\sqrt{x-3}+2)+ \frac{\delta}{2}(x-3), \;\; 3\le x \le 3.4... \\  ...  \end{matrix}  \end{equation} Where $\delta$ appears to be the Feigenbaum constant 4.669201609102990671853203821578...