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zeros of analytic functions

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1: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
§3.8(v) Zeros of Analytic Functions
Newton’s rule is the most frequently used iterative process for accurate computation of real or complex zeros of analytic functions f ( z ) . …
§3.8(vi) Conditioning of Zeros
Corresponding numerical factors in this example for other zeros and other values of j are obtained in Gautschi (1984, §4). …
2: 4.14 Definitions and Periodicity
4.14.7 cot z = cos z sin z = 1 tan z .
3: 5.2 Definitions
5.2.1 Γ ( z ) = 0 e - t t z - 1 d t , z > 0 .
4: 1.10 Functions of a Complex Variable
An analytic function f ( z ) has a zero of order (or multiplicity) m ( 1 ) at z 0 if the first nonzero coefficient in its Taylor series at z 0 is that of ( z - z 0 ) m . … … then the product n = 1 ( 1 + a n ( z ) ) converges uniformly to an analytic function p ( z ) in D , and p ( z ) = 0 only when at least one of the factors 1 + a n ( z ) is zero in D . …
5: 4.28 Definitions and Periodicity
Periodicity and Zeros
6: 10.21 Zeros
§10.21(ii) Analytic Properties
7: Bibliography C
  • J. A. Cochran (1966a) The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.
  • 8: 4.2 Definitions
    §4.2(iii) The Exponential Function
    9: 23.2 Definitions and Periodic Properties
    §23.2(i) Lattices
    §23.2(ii) Weierstrass Elliptic Functions
    The function σ ( z ) is entire and odd, with simple zeros at the lattice points. … …
    10: 2.4 Contour Integrals
    Now assume that c > 0 and we are given a function Q ( z ) that is both analytic and has the expansion … Assume that p ( t ) and q ( t ) are analytic on an open domain T that contains 𝒫 , with the possible exceptions of t = a and t = b . …
  • (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 .

  • in which z is a large real or complex parameter, p ( α , t ) and q ( α , t ) are analytic functions of t and continuous in t and a second parameter α . … The function f ( α , w ) is analytic at w = w 1 ( α ) and w = w 2 ( α ) when α α ^ , and at the confluence of these points when α = α ^ . …