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\end{bmatrix} \]  This leaves us then with \[\tilde{A}(z)=\tilde{A}(0)\left( cos \beta_c z -\frac{i \delta}{\beta_c} sin \beta_c z \right) e^{i \delta z} \]  \[\tilde{B}(z)=\tilde{A}(0)\left(\frac{i \kappa_{ba}}{\beta_c} sin \beta_c z \right) e^{-i \delta z} \]  Looking back at my slides I wrote that $[\tilde{B}(z)=\tilde{B}(0)\left(\frac{i $\tilde{B}(z)=\tilde{B}(0)\left(\frac{i  \kappa_{ba}}{\beta_c} sin \beta_c z \right) e^{-i \delta z} $, which given my initial conditions would be wrong since $\tilde{B}(0)=0$. Thankfully, this error doesn't propagate through the rest of the presentation as that error only shows up on slide 8 and the rest of the figures I made are still accurate.