= Bounded vertical modes of a sech-squared magnetized disk
{title2=$F_n=P_n(\tanh(z/H)),\quad k_n^2H^2=n(n+1)$}
For density proportional to $\operatorname{sech}^2(z/H)$, the magnetic vertical equation $F''+k^2\operatorname{sech}^2(z/H)F=0$ becomes the <Legendre differential equation> under $\xi=\tanh(z/H)$. Bounded solutions at both surfaces are <Legendre polynomials>, with $k_n=\sqrt{n(n+1)}/H$. The $n=0$ constant function has no magnetically coupled finite wavenumber. The smallest MRI-relevant eigenvalue is $k_1^2=2/H^2$, which makes sufficiently strong <magnetic tension> stabilize all allowed modes.
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