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21: 1.13 Differential Equations
§1.13(vi) Singularities
For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. … on a finite interval [ a , b ] , this is then a regular Sturm-Liouville system. … A regular Sturm-Liouville system will only have solutions for certain (real) values of λ , these are eigenvalues. … For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, λ ; (ii) the corresponding (real) eigenfunctions, u ( x ) and w ( t ) , have the same number of zeros, also called nodes, for t ( 0 , c ) as for x ( a , b ) ; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …
22: Bibliography L
  • W. Lay and S. Yu. Slavyanov (1999) Heun’s equation with nearby singularities. Proc. Roy. Soc. London Ser. A 455, pp. 4347–4361.
  • S. Lewanowicz (1991) Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.
  • X. Li, X. Shi, and J. Zhang (1991) Generalized Riemann ζ -function regularization and Casimir energy for a piecewise uniform string. Phys. Rev. D 44 (2), pp. 560–562.
  • 23: Bibliography
  • J. Abad and J. Sesma (1995) Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.
  • M. Abramowitz (1954) Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.
  • A. R. Ahmadi and S. E. Widnall (1985) Unsteady lifting-line theory as a singular-perturbation problem. J. Fluid Mech 153, pp. 59–81.
  • H. H. Aly, H. J. W. Müller-Kirsten, and N. Vahedi-Faridi (1975) Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.
  • J. Avron and B. Simon (1982) Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.
  • 24: 18.39 Applications in the Physical Sciences
    Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. …
    See accompanying text
    Figure 18.39.2: Coulomb–Pollaczek weight functions, x [ 1 , 1 ] , (18.39.50) for s = 10 , l = 0 , and Z = ± 1 . For Z = + 1 the weight function, red curve, has an essential singularity at x = 1 , as all derivatives vanish as x 1 + ; the green curve is 1 x w CP ( y ) d y , to be compared with its histogram approximation in §18.40(ii). For Z = 1 the weight function, blue curve, is non-zero at x = 1 , but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as k , x k 1 . Magnify
    The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as x 1 . … See Yamani and Fishman (1975) for L 2 for expansions of both the regular and irregular spherical Bessel functions, which are the Pollaczeks with a = Z = 0 , and Coulomb functions for fixed l , Broad and Reinhardt (1976) for a many particle example, and the overview of Alhaidari et al. (2008). …