Regge poles
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11: 1.10 Functions of a Complex Variable
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►Lastly, if for infinitely many negative , then is an isolated essential singularity.
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►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function.
If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles.
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►If the singularities within are poles and is analytic and nonvanishing on , then
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►each location again being counted with multiplicity equal to that of the corresponding zero or pole.
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12: 16.17 Definition
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►where the integration path separates the poles of the factors from those of the factors .
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(ii)
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(iii)
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is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all () if , and for if .
is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all if , and for if .
13: 5.2 Definitions
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►It is a meromorphic function with no zeros, and with simple poles of residue at .
… is meromorphic with simple poles of residue at .
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14: 32.11 Asymptotic Approximations for Real Variables
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►Next, for given initial conditions and , with real, has at least one pole on the real axis.
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►If , then has a pole at a finite point , dependent on , and
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►then has no poles on the real axis.
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►and has no poles on the real axis.
►Lastly if , then has a simple pole on the real axis, whose location is dependent on .
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15: 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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16: 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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17: 8.15 Sums
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8.15.2
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18: 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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19: 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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