I'm working on solving a 3d particle in a well for quantum mechanics.
I have the transcendental equation $$ - \cot(z) = \sqrt{(z_0/z)^2-1} $$
A graph of this looks like
Plot[{Sqrt[(5*\[Pi]/z)^2 - 1], -Cot[z]}, {z, 0, 5*\[Pi]}]
A solution manual for Griffiths QM book states that the solutions ONLY occur at multiples of pi, and supplies the following graph
why are those other non-intervals of pi intersections not considered?
On
Consider that you are looking for the zero's of function $$f(z)=\sqrt{\frac{a^2}{z^2}-1}+\cot (z)$$ You cannot have any solution if $z \gt a$.
Concerning the possible roots, using Taylor expansion around $z=k \pi$ and series reversion, they are given by $$z_{(k)}=k \pi \left(1-\frac{1}{\left(a^2-\pi ^2 k^2\right)^{1/2}}+\frac{\pi ^2 k^2}{2 \left(a^2-\pi ^2 k^2\right)^{3/2}}+\frac{a^2}{\left(a^2-\pi ^2 k^2\right)^2} +\cdots\right)$$
If that's what the manual says, it is wrong. Multiples of $\pi$ are not solutions, as $\cot$ is not defined there. Rather, there is one solution in each interval between two multiples of $\pi$.