isolated%20essential
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1: 1.10 Functions of a Complex Variable
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►Then is an isolated singularity of .
…Lastly, if for infinitely many negative , then is an isolated essential singularity.
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►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in with at most one exception.
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2: 32.2 Differential Equations
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►be a nonlinear second-order differential equation in which is a rational function of and , and is locally analytic in , 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.
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3: 20 Theta Functions
Chapter 20 Theta Functions
…4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where is continuous, with convergence to if is an isolated point of discontinuity.
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►An essential feature of such symmetric operators is that their eigenvalues are real, and eigenfunctions
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►In unusual cases , even for all , such as in the case of the Schrödinger–Coulomb problem () discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at , implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66).
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►If then is essentially self-adjoint and if then is self-adjoint.
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5: 14.32 Methods of Computation
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►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.
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6: Bibliography K
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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).
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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7: 34.12 Physical Applications
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►The angular momentum coupling coefficients (, , and symbols) are essential in the fields of nuclear, atomic, and molecular physics.
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8: 18.40 Methods of Computation
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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►This is a challenging case as the desired on has an essential singularity at .
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9: 8 Incomplete Gamma and Related
Functions
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10: 28 Mathieu Functions and Hill’s Equation
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