fix typo in equation (#2184)

doc/docs/Python_Tutorials/Local_Density_of_States.md

@@ -223,7 +223,7 @@ if __name__ == '__main__':
 Square Box with a Small Opening
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-We consider the simple example of a 2D perfect-metal $a$x$a$ cavity of finite thickness 0.1$a$, with a small notch of width $w$ on one side that allows the modes to escape. The nice thing about this example is that in the absence of the notch, the lowest-frequency $E_z$-polarized mode is known analytically to be $E_z^{(1)}=\frac{4}{a^2}\sin(\pi x/a)\sin(\pi \gamma/a)$, with a frequency $\omega^{(1)}=\sqrt{2}\pi c/a$ and modal volume $V^{(1)}=a^2/4$. The notch slightly perturbs this solution, but more importantly the opening allows the confined mode to radiate out into the surrounding air, yielding a finite $Q$. For $w \ll a$, this radiative escape occurs via an evanescent (sub-cutoff) mode of the channel waveguide formed by the notch, and it follows from inspection of the evanescent decay rate $\sqrt{(\pi/\omega)^2-(\omega^{(1)})^2}/c$ that the lifetime scales asymptotically as $Q^{(1)} \sim e^{\\#/\omega}$ for some coefficient \#.
+We consider the simple example of a 2D perfect-metal $a$x$a$ cavity of finite thickness 0.1$a$, with a small notch of width $w$ on one side that allows the modes to escape. The nice thing about this example is that in the absence of the notch, the lowest-frequency $E_z$-polarized mode is known analytically to be $E_z^{(1)}=\frac{4}{a^2}\sin(\pi x/a)\sin(\pi \gamma/a)$, with a frequency $\omega^{(1)}=\sqrt{2}\pi c/a$ and modal volume $V^{(1)}=a^2/4$. The notch slightly perturbs this solution, but more importantly the opening allows the confined mode to radiate out into the surrounding air, yielding a finite $Q$. For $w \ll a$, this radiative escape occurs via an evanescent (sub-cutoff) mode of the channel waveguide formed by the notch, and it follows from inspection of the evanescent decay rate $\sqrt{(\pi/\omega)^2-(\omega^{(1)})^2}/c$ that the lifetime scales asymptotically as $Q^{(1)} \sim e^{\#/\omega}$ for some coefficient \#.
 
 We will validate both this prediction and the expression for the LDOS shown above by computing the LDOS at the center of the cavity, the point of peak $|\vec{E}|$, in two ways. First, we compute the LDOS directly from the power radiated by a dipole, Fourier-transforming the result of a pulse using the `dft_ldos` command. Second, we compute the cavity mode and its lifetime $Q$ using Harminv and then compute the LDOS by the Purcell formula shown above. The latter technique is much more efficient for high Q (small $w$), since one must run the simulation for a very long time to directly accumulate the Fourier transform of a slowly-decaying mode. The two calculations, we will demonstrate, agree to within discretization error, verifying the LDOS analysis above, and $Q/V$ is asymptotically linear on a semilog scale versus $1/w$ as predicted.