\begin{align} f(\theta)& = \log \frac{1}{1-\theta}-\frac{\theta}{2-\theta}+\frac{\theta^{2}}{(2-\theta)^{2}} \\ & = \log (1-\theta)+\frac{\theta}{2-\theta} \end{align} How are the two equations above equal to each other, i.e. reduce the first into the second?
How can one term be outside of the logarithm on the second line, whereas all 3 terms fall inside the logarithm on the first line? Actually, I'm not even sure if all the terms in the first line are supposed to all fall inside the logarithm, no brackets were given, so it could be either or
Source of first equation:
- Formula 22 in Mercier 2005
Source of second equation:
- Formula 4.4 in Kumar 2011
If they don't equal each other like how the second author claims, which author made a mistake?
Edit
I found yet another equality presented by the same second author. This time,
- Formula 3.41 of Kumar 2014:
$$ f(\theta) = \log (1-\theta)+\frac{2 \theta(1-\theta)}{(2-\theta)^{2}} $$
The first line is not equal to the second line:
$$f(\theta) = \log \left( \frac{1}{1-\theta} \right)-\frac{\theta}{2-\theta}+\frac{\theta^{2}}{(2-\theta)^{2}}$$ $$= \log( (1 - \theta)^{-1}) - \frac{\theta (2 - \theta)}{(2 - \theta)^2} + \frac{\theta^{2}}{(2-\theta)^{2}}$$ $$= -\log(1 - \theta) + \frac{-(2\theta - \theta^2) + \theta^2}{(2 - \theta)^2}$$ $$= -\log(1 - \theta) + \frac{2 \theta^2 - 2 \theta}{(2 - \theta)^2}$$ $$= -\log(1 - \theta) + \frac{(2\theta)(\theta - 1)}{(2 - \theta)^2}$$