\[\theta_{n,m}=\phi_{m}+\left(\left(\alpha^{2}-\tfrac{1}{4}\right)\frac{1-\phi_{%
m}\cot\phi_{m}}{2\phi_{m}}-\tfrac{1}{4}(\alpha^{2}-\beta^{2})\tan\left(\tfrac{%
1}{2}\phi_{m}\right)\right)\frac{1}{\rho^{2}}+\phi_{m}^{2}O\left(\frac{1}{\rho%
^{3}}\right),\]