Geodesic
Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 11, pp. 1880-1897

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A method for computing the eigenvalues $\lambda_{mn}(b,c)$ and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters $b$ and $c$. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain $b_s$ and $c_s$ are second-order branch points for $\lambda_{mn}(b,c)$ with different indices $n_1$ and $n_2$, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite–Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points $b_s$ and $c_s$ and the double eigenvalues to high accuracy. A large number of these singular points are calculated. 
@article{ZVMMF_2007_47_11_a6,
     author = {S. L. Skorokhodov and D. V. Khristoforov},
     title = {Calculating the branch points of the eigenvalues of the {Coulomb} spheroidal wave equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1880--1897},
     publisher = {mathdoc},
     volume = {47},
     number = {11},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_11_a6/}
}
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S. L. Skorokhodov; D. V. Khristoforov. Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 11, pp. 1880-1897. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_11_a6/