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 . … ►§3.8(vi) Conditioning of Zeros
… ►Corresponding numerical factors in this example for other zeros and other values of are obtained in Gautschi (1984, §4). …2: 4.14 Definitions and Periodicity
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4.14.7
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3: 5.2 Definitions
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5.2.1
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4: 1.10 Functions of a Complex Variable
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►An analytic function
has a zero of order (or multiplicity) () at if the first nonzero coefficient in its Taylor series at is that of .
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►then the product converges uniformly to an analytic function
in , and only when at least one of the factors is zero in .
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5: 4.28 Definitions and Periodicity
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Periodicity and Zeros
…6: 10.21 Zeros
7: Bibliography C
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The analyticity of cross-product Bessel function zeros.
Proc. Cambridge Philos. Soc. 62, pp. 215–226.
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8: 4.2 Definitions
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§4.2(iii) The Exponential Function
…9: 23.2 Definitions and Periodic Properties
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§23.2(i) Lattices
… ► … ►§23.2(ii) Weierstrass Elliptic Functions
… ►The function is entire and odd, with simple zeros at the lattice points. … …10: 2.4 Contour Integrals
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►Now assume that and we are given a function
that is both analytic and has the expansion
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►Assume that and are analytic on an open domain that contains , with the possible exceptions of and .
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(a)
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►in which is a large real or complex parameter, and are analytic functions of and continuous in and a second parameter .
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►The function
is analytic at and when , and at the confluence of these points when .
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In a neighborhood of
2.4.11
with , , , and the branches of and continuous and constructed with as along .