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11: 31.7 Relations to Other Functions
Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities ζ = K , K + i K , and i K , where K and K are related to k as in §19.2(ii).
12: 15.10 Hypergeometric Differential Equation
They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity. … (a) If c equals n = 1 , 2 , 3 , , and a = 1 , 2 , , n - 1 , then fundamental solutions in the neighborhood of z = 0 are given by (15.10.2) with the interpretation (15.2.5) for f 2 ( z ) . (b) If c equals n = 1 , 2 , 3 , , and a 1 , 2 , , n - 1 , then fundamental solutions in the neighborhood of z = 0 are given by F ( a , b ; n ; z ) and … (c) If the parameter c in the differential equation equals 2 - n = 0 , - 1 , - 2 , , then fundamental solutions in the neighborhood of z = 0 are given by z n - 1 times those in (a) and (b), with a and b replaced throughout by a + n - 1 and b + n - 1 , respectively. (d) If a + b + 1 - c equals n = 1 , 2 , 3 , , or 2 - n = 0 , - 1 , - 2 , , then fundamental solutions in the neighborhood of z = 1 are given by those in (a), (b), and (c) with z replaced by 1 - z . …
13: Bibliography
  • V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
  • 14: 1.13 Differential Equations
    For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
    15: 3.7 Ordinary Differential Equations
    For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
    16: 33.14 Definitions and Basic Properties
    This includes ϵ = 0 , hence f ( ϵ , ; r ) can be expanded in a convergent power series in ϵ in a neighborhood of ϵ = 0 33.20(ii)). …
    17: 1.5 Calculus of Two or More Variables
    Implicit Function Theorem
    If F ( x , y ) is continuously differentiable, F ( a , b ) = 0 , and F / y 0 at ( a , b ) , then in a neighborhood of ( a , b ) , that is, an open disk centered at a , b , the equation F ( x , y ) = 0 defines a continuously differentiable function y = g ( x ) such that F ( x , g ( x ) ) = 0 , b = g ( a ) , and g ( x ) = - F x / F y . …
    18: 1.14 Integral Transforms
    Suppose that f ( t ) is absolutely integrable on ( - , ) and of bounded variation in a neighborhood of t = u 1.4(v)). … If f ( t ) is absolutely integrable on [ 0 , ) and of bounded variation (§1.4(v)) in a neighborhood of t = u , then … Suppose the integral (1.14.32) is absolutely convergent on the line s = σ and f ( x ) is of bounded variation in a neighborhood of x = u . …
    19: 2.4 Contour Integrals
  • (a)

    In a neighborhood of a

    2.4.11
    p ( t ) = p ( a ) + s = 0 p s ( t - a ) s + μ ,
    q ( t ) = s = 0 q s ( t - a ) s + λ - 1 ,

    with λ > 0 , μ > 0 , p 0 0 , and the branches of ( t - a ) λ and ( t - a ) μ continuous and constructed with ph ( t - a ) ω as t a along 𝒫 .

  • 20: 35.7 Gaussian Hypergeometric Function of Matrix Argument
    Let f : Ω (a) be orthogonally invariant, so that f ( T ) is a symmetric function of t 1 , , t m , the eigenvalues of the matrix argument T Ω ; (b) be analytic in t 1 , , t m in a neighborhood of T = 0 ; (c) satisfy f ( 0 ) = 1 . …