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I found the following equation in a paper: $$\alpha=\sum\limits_{i=l}^2\sum\limits_{l=1}^2(-1)^l\frac{\alpha_i\left[hf-E_g(T)+(-1)^lk\theta_i\right]}{\left(\exp\left[(-1)^l\frac{\theta_i}{T}\right]-1\right)}$$ Now I am wondering how to interpret this equation regarding $i$:

Does it rather look (when written with all sum terms) like $$ \begin{split} \alpha&=\left(\frac{\alpha_1\left[hf-E_g(T)+k\theta_1\right]}{\left(\exp\left[\frac{\theta_1}{T}\right]-1\right)}-\frac{\alpha_1\left[hf-E_g(T)-k\theta_1\right]}{\left(\exp\left[-\frac{\theta_1}{T}\right]-1\right)}\right)\\ &+\left(\frac{\alpha_2\left[hf-E_g(T)+k\theta_2\right]}{\left(\exp\left[\frac{\theta_i}{T}\right]-1\right)}+\frac{\alpha_2\left[hf-E_g(T)-k\theta_2\right]}{\left(\exp\left[-\frac{\theta_2}{T}\right]-1\right)}\right) \end{split}$$ or $$ \begin{split} \alpha&=\left(\frac{\alpha_1\left[hf-E_g(T)+k\theta_1\right]}{\left(\exp\left[\frac{\theta_1}{T}\right]-1\right)}-\frac{\alpha_1\left[hf-E_g(T)-k\theta_1\right]}{\left(\exp\left[-\frac{\theta_1}{T}\right]-1\right)}\right)\\ &+\left(\frac{\alpha_2\left[hf-E_g(T)+k\theta_2\right]}{\left(\exp\left[\frac{\theta_2}{T}\right]-1\right)}\right) \end{split} $$ Which representation is correct? Or is none of them?

arc_lupus
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  • It looks like a typo in the paper as it doesn't make sense in the current form. $i$ can't depend on $l$ because $l$ is a temporary index. My guess is that the two summation signs should be swapped and then the latter expression with three terms is correct. – Benedict W. J. Irwin Oct 16 '17 at 13:43

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