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21: 22.10 Maclaurin Series
The radius of convergence is the distance to the origin from the nearest pole in the complex k -plane in the case of (22.10.4)–(22.10.6), or complex k -plane in the case of (22.10.7)–(22.10.9); see §22.17. …
22: 25.2 Definition and Expansions
It is a meromorphic function whose only singularity in is a simple pole at s = 1 , with residue 1. …
25.2.4 ζ ( s ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( s - 1 ) n ,
23: Bibliography N
  • J. J. Nestor (1984) 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.
  • 24: 2.8 Differential Equations with a Parameter
    §2.8(iv) Case III: Simple Pole
    More generally, g ( z ) can have a simple or double pole at z 0 . (In the case of the double pole the order of the approximating Bessel functions is fixed but no longer 1 / ( λ + 2 ) .) … For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …
    25: 13.4 Integral Representations
    where the contour of integration separates the poles of Γ ( a + t ) from those of Γ ( - t ) . … where the contour of integration separates the poles of Γ ( a + t ) Γ ( 1 + a - b + t ) from those of Γ ( - t ) . …where the contour of integration passes all the poles of Γ ( b - 1 + t ) Γ ( t ) on the right-hand side. …
    26: 13.16 Integral Representations
    where the contour of integration separates the poles of Γ ( t - κ ) from those of Γ ( 1 2 + μ - t ) . … where the contour of integration separates the poles of Γ ( 1 2 + μ + t ) Γ ( 1 2 - μ + t ) from those of Γ ( - κ - t ) . …where the contour of integration passes all the poles of Γ ( 1 2 + μ + t ) Γ ( 1 2 - μ + t ) on the right-hand side.
    27: 32.2 Differential Equations
    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. …
    28: 19.14 Reduction of General Elliptic Integrals
    It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …
    29: 22.3 Graphics
    See accompanying text
    Figure 22.3.26: Density plot of | sn ( 5 , k ) | as a function of complex k 2 , - 10 ( k 2 ) 20 , - 10 ( k 2 ) 10 . …White spots correspond to poles. Magnify
    See accompanying text
    Figure 22.3.27: Density plot of | sn ( 10 , k ) | as a function of complex k 2 , - 10 ( k 2 ) 20 , - 10 ( k 2 ) 10 . …White spots correspond to poles. Magnify
    See accompanying text
    Figure 22.3.28: Density plot of | sn ( 20 , k ) | as a function of complex k 2 , - 10 ( k 2 ) 20 , - 10 ( k 2 ) 10 . …White spots correspond to poles. Magnify
    See accompanying text
    Figure 22.3.29: Density plot of | sn ( 30 , k ) | as a function of complex k 2 , - 10 ( k 2 ) 20 , - 10 ( k 2 ) 10 . …White spots correspond to poles. Magnify
    30: 25.15 Dirichlet L -functions
    For the principal character χ 1 ( mod k ) , L ( s , χ 1 ) is analytic everywhere except for a simple pole at s = 1 with residue ϕ ( k ) / k , where ϕ ( k ) is Euler’s totient function (§27.2). …