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11: 4.14 Definitions and Periodicity
The functions tan z , csc z , sec z , and cot z are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …
12: 7.20 Mathematical Applications
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). …
13: 8.15 Sums
8.15.2 a k = 1 ( e 2 π i k ( z + h ) ( 2 π i k ) a + 1 Γ ( a , 2 π i k z ) + e - 2 π i k ( z + h ) ( - 2 π i k ) a + 1 Γ ( a , - 2 π i k z ) ) = ζ ( - a , z + h ) + z a + 1 a + 1 + ( h - 1 2 ) z a , h [ 0 , 1 ] .
14: 23.2 Definitions and Periodic Properties
( z ) and ζ ( z ) are meromorphic functions with poles at the lattice points. …The poles of ( z ) are double with residue 0 ; the poles of ζ ( z ) are simple with residue 1 . … …
15: Bibliography D
  • B. Deconinck and H. Segur (2000) Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.
  • T. M. Dunster (1990b) 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.
  • T. M. Dunster (1994b) 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.
  • T. M. Dunster (2004) 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.
  • 16: 33.22 Particle Scattering and Atomic and Molecular Spectra
    §33.22(vii) Complex Variables and Parameters
  • Searches for resonances as poles of the S -matrix in the complex half-plane k < 0 . 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
    where the contour separates the poles of Γ ( t ) from those of Γ ( 1 2 + a - 2 t ) . … where the contour separates the poles of Γ ( t ) from those of Γ ( 1 2 - a - 2 t ) . …
    18: 16.11 Asymptotic Expansions
    It may be observed that H p , q ( z ) represents the sum of the residues of the poles of the integrand in (16.5.1) at s = - a j , - a j - 1 , , j = 1 , , p , provided that these poles are all simple, that is, no two of the a j differ by an integer. (If this condition is violated, then the definition of H p , q ( z ) has to be modified so that the residues are those associated with the multiple poles. …
    19: 14.21 Definitions and Basic Properties
    P ν ± μ ( z ) and Q ν μ ( z ) exist for all values of ν , μ , and z , except possibly z = ± 1 and , which are branch points (or poles) of the functions, in general. …
    20: 15.3 Graphics
    See accompanying text
    Figure 15.3.6: F ( - 3 , 3 5 ; u + i v ; 1 2 ) , - 6 u 2 , - 2 v 2 . (With c = u + i v the only poles occur at c = 0 , - 1 , - 2 ; compare §15.2(ii).) Magnify 3D Help