recreate missing LaTeX files for image-only equations

\documentclass[12pt]{article}

\pagestyle{empty}

\begin{document}

$$

  E_{a} = K_2\left(\theta - \theta_0\right)^2 + K_3\left(\theta - \theta_0\right)^3 + K_4\left(\theta - \theta_0\right)^4 + K_5\left(\theta - \theta_0\right)^5 + K_6\left(\theta - \theta_0\right)^6

$$

\end{document}

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\documentclass[12pt]{article}

\pagestyle{empty}

\begin{document}

$$

  E = K \left[ 1 + \cos(n\theta - \theta_0)\right]

$$

\end{document}

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\documentclass[12pt]{article}

\pagestyle{empty}

\begin{document}

$$

  E = K \left(\theta - \theta_0\right)^2

$$

\end{document}

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\documentclass[12pt]{article}

\pagestyle{empty}

\begin{document}

\begin{eqnarray*}

  F_i^u & = & \sum_t^{N_t}\alpha_t \cdot \exp\left[-\frac{\left(d_{i,t}^u\right)^2}{2l^2}\right] \\

  d_{i,t}^u & = & \left|\left| V_i^u(\eta) - V_t^u(\eta) \right|\right| \\

  V_i^u(\eta) & = & \sum_{j \neq i}\frac{r^u_{ij}}{r_{ij}} \cdot e^{-\left(\frac{r_{ij}}{\eta} \right)^2} \cdot f_d\left(r_{ij}\right) \\

  f_d\left(r_{ij}\right) & = & \frac{1}{2} \left[\cos\left(\frac{\pi r_{ij}}{R_c}\right) + 1 \right]

\end{eqnarray*}

\end{document}

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\documentclass[12pt]{article}

\begin{document}

\pagestyle{empty}

\begin{eqnarray*}

  E = A e^{-\kappa r} - \frac{C}{r^6} \cdot \frac{1}{1 + D r^{14}} \qquad r < r_c \\