poles
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11: 4.14 Definitions and Periodicity
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►The functions , , , and are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7).
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12: 7.20 Mathematical Applications
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►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951).
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13: 8.15 Sums
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8.15.2
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14: 23.2 Definitions and Periodic Properties
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and are meromorphic functions with poles at the lattice points.
…The poles of are double with residue ; the poles of are simple with residue .
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15: Bibliography D
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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation.
Math. Phys. Anal. Geom. 3 (1), pp. 49–74.
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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point.
SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane.
SIAM J. Math. Anal. 25 (2), pp. 322–353.
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Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions.
Stud. Appl. Math. 113 (3), pp. 245–270.
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16: 33.22 Particle Scattering and Atomic and Molecular Spectra
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§33.22(vii) Complex Variables and Parameters
… ►Searches for resonances as poles of the -matrix in the complex half-plane . See for example Csótó and Hale (1997).
Regge poles at complex values of . See for example Takemasa et al. (1979).
17: 12.5 Integral Representations
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►where the contour separates the poles of from those of .
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►where the contour separates the poles of from those of .
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18: 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.
19: 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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20: 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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