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31: Bibliography
  • 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.
  • V. I. Arnol’d, S. M. Guseĭn-Zade, and A. N. Varchenko (1988) Singularities of Differentiable Maps. Vol. II. Birkhäuser, Boston-Berlin.
  • V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups A k , D k , E k and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
  • 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.
  • 32: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    Should q ( x ) be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). … … Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies λ res i Γ res / 2 corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to 1 / Γ res . For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. … The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. …
    33: 15.19 Methods of Computation
    However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. …
    34: 21.7 Riemann Surfaces
    This compact curve may have singular points, that is, points at which the gradient of P ~ vanishes. Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way.On this surface, we choose 2 g cycles (that is, closed oriented curves, each with at most a finite number of singular points) a j , b j , j = 1 , 2 , , g , such that their intersection indices satisfy … Thus the differentials ω j , j = 1 , 2 , , g have no singularities on Γ . …
    35: 33.2 Definitions and Basic Properties
    §33.2(i) Coulomb Wave Equation
    This differential equation has a regular singularity at ρ = 0 with indices + 1 and , and an irregular singularity of rank 1 at ρ = (§§2.7(i), 2.7(ii)). …
    36: 33.14 Definitions and Basic Properties
    §33.14(i) Coulomb Wave Equation
    Again, there is a regular singularity at r = 0 with indices + 1 and , and an irregular singularity of rank 1 at r = . …
    37: 31.11 Expansions in Series of Hypergeometric Functions
    and (31.11.1) converges to (31.3.10) outside the ellipse in the z -plane with foci at 0, 1, and passing through the third finite singularity at z = a . Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function. … The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities z = 0 and z = 1 . …
    38: Bibliography S
  • A. Sidi (2004) Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective. Numer. Math. 98 (2), pp. 371–387.
  • A. Sidi (2010) A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.
  • A. Sidi (2012a) Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp. 81 (280), pp. 2159–2173.
  • A. Sidi (2012b) Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities. Constr. Approx. 36 (3), pp. 331–352.
  • B. Simon (1995) Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.
  • 39: 15.10 Hypergeometric Differential Equation
    It has regular singularities at z = 0 , 1 , , with corresponding exponent pairs { 0 , 1 c } , { 0 , c a b } , { a , b } , respectively. …They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity.
    Singularity z = 0
    Singularity z = 1
    Singularity z =
    40: 2.10 Sums and Sequences
    For extensions of the Euler–Maclaurin formula to functions f ( x ) with singularities at x = a or x = n (or both) see Sidi (2004, 2012b, 2012a). … However, if r is finite and f ( z ) has algebraic or logarithmic singularities on | z | = r , then Darboux’s method is usually easier to apply. … in the neighborhood of each singularity z j , again with σ j > 0 . … The singularities of f ( z ) on the unit circle are branch points at z = e ± i α . … For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005). …