Regge poles
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21: 13.16 Integral Representations
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►where the contour of integration separates the poles of from those of .
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►where the contour of integration separates the poles of from those of .
…where the contour of integration passes all the poles of on the right-hand side.
22: 16.11 Asymptotic Expansions
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►It may be observed that represents the sum of the residues of the poles of the integrand in (16.5.1) at , , provided that these poles are all simple, that is, no two of the differ by an integer.
(If this condition is violated, then the definition of has to be modified so that the residues are those associated with the multiple poles.
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23: 14.21 Definitions and Basic Properties
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and exist for all values of , , and , except possibly and , which are branch points (or poles) of the functions, in general.
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24: 15.3 Graphics
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25: 22.10 Maclaurin Series
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►The radius of convergence is the distance to the origin from the nearest pole in the complex -plane in the case of (22.10.4)–(22.10.6), or complex -plane in the case of (22.10.7)–(22.10.9); see §22.17.
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26: 18.40 Methods of Computation
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►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
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27: 25.2 Definition and Expansions
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►It is a meromorphic function whose only singularity in is a simple pole at , with residue 1.
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25.2.4
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28: Bibliography N
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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole.
Ph.D. Thesis, University of Maryland, College Park, MD.
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29: 2.8 Differential Equations with a Parameter
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