analyticity
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21: 15.17 Mathematical Applications
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►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group.
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22: 10.72 Mathematical Applications
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)).
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►The number can also be replaced by any real constant
in the sense that
is analytic and nonvanishing at ; moreover, is permitted to have a single or double pole at .
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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►Assume that whether or not , is analytic at .
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23: 31.4 Solutions Analytic at Two Singularities: Heun Functions
§31.4 Solutions Analytic at Two Singularities: Heun Functions
►For an infinite set of discrete values , , of the accessory parameter , the function is analytic at , and hence also throughout the disk . … ►with , denotes a set of solutions of (31.2.1), each of which is analytic at and . …24: 15.2 Definitions and Analytical Properties
§15.2 Definitions and Analytical Properties
… ►on the disk , and by analytic continuation elsewhere. … ►again with analytic continuation for other values of , and with the principal branch defined in a similar way. … ►§15.2(ii) Analytic Properties
… ►The right-hand side can be seen as an analytical continuation for the left-hand side when approaches . …25: 16.2 Definition and Analytic Properties
§16.2 Definition and Analytic Properties
… ►§16.2(ii) Case
… ►§16.2(iii) Case
… ►§16.2(iv) Case
… ►§16.2(v) Behavior with Respect to Parameters
…26: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
… ►This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: … ►§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
…27: 35.2 Laplace Transform
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►where the integration variable ranges over the space .
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►Then (35.2.1) converges absolutely on the region , and is a complex analytic function of all elements of .
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28: 3.12 Mathematical Constants
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►can be defined analytically in numerous ways, for example,
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