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1: 1.10 Functions of a Complex Variable
Then z = z 0 is an isolated singularity of f ( z ) . …Lastly, if a n 0 for infinitely many negative n , then z 0 is an isolated essential singularity. … In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in with at most one exception. …
2: 32.2 Differential Equations
be a nonlinear second-order differential equation in which F is a rational function of w and d w / d z , and is locally analytic in z , that is, analytic except for isolated singularities in . …An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. …
3: 20 Theta Functions
Chapter 20 Theta Functions
4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where f ( x ) is continuous, with convergence to ( f ( x 0 ) + f ( x 0 + ) ) / 2 if x 0 is an isolated point of discontinuity. … An essential feature of such symmetric operators is that their eigenvalues λ are real, and eigenfunctions … In unusual cases N = , even for all , such as in the case of the Schrödinger–Coulomb problem ( V = r 1 ) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at λ = 0 , implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … If T T = T then T is essentially self-adjoint and if T = T then T is self-adjoint. …
5: 14.32 Methods of Computation
Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. …
6: Bibliography K
  • A. A. Kapaev (1991) Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187, pp. 139–170 (Russian).
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • 7: 34.12 Physical Applications
    The angular momentum coupling coefficients ( 3 j , 6 j , and 9 j symbols) are essential in the fields of nuclear, atomic, and molecular physics. …
    8: 18.40 Methods of Computation
    Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . … This is a challenging case as the desired w RCP ( x ) on [ 1 , 1 ] has an essential singularity at x = 1 . …
    9: 8 Incomplete Gamma and Related
    Functions
    10: 28 Mathieu Functions and Hill’s Equation