👩🏼‍🏫 Outreach in mathematics

\begin{align*}
		&\sum_{z \in \mathbb{C} \setminus \{-1\}: z^\alpha=-1} \text{Res}(J_{\alpha}(t, \cdot), z) 
		= \frac{1}{2\pi i} \oint_{\Gamma_{\gamma, \delta, R}} J_{\alpha}(t, z) \, dz 
		= \frac{1}{2\pi i}\int_{\textit{Br}(\gamma, R)} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C^+} J_{\alpha}(t, z) \, dz \\
		&\quad \hspace{1.5cm} + \frac{1}{2\pi i}\int_{C_R^+} J_{\alpha}(t, z) \, dz - \frac{1}{2\pi i}\int_{\textit{H}(\delta, \frac{1}{R})} J_{\alpha}(t, z) \, dz  + \frac{1}{2\pi i}\int_{C_R^-} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C^-} J_{\alpha}(t, z) \, dz.
\end{align*}