branch point
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1: 28.7 Analytic Continuation of Eigenvalues
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►The only singularities are algebraic branch points, with and finite at these points.
The number of branch points is infinite, but countable, and there are no finite limit points.
…The branch points are called the exceptional values, and the other points normal values.
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►For a visualization of the first branch point of and see Figure 28.7.1.
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2: Mark J. Ablowitz
3: 1.10 Functions of a Complex Variable
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►Then is a branch point of .
For example, is a branch point of .
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►(a) By introducing appropriate cuts from the branch points and restricting to be single-valued in the cut plane (or domain).
►(b) By specifying the value of at a point
(not a branch point), and requiring to be continuous on any path that begins at and does not pass through any branch points or other singularities of .
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4: 14.24 Analytic Continuation
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►Next, let and denote the branches obtained from the principal branches by encircling the branch point
(but not the branch point
) times in the positive sense.
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5: 19.14 Reduction of General Elliptic Integrals
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►A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges.
If no such branch point is accessible from the interval of integration (for example, if the integrand is and the interval is [1,2]), then no method using this assumption succeeds.
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6: 21.7 Riemann Surfaces
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►The zeros , of specify the finite branch points
, that is, points at which , on the Riemann surface.
Denote the set of all branch points by .
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21.7.17
7: 4.37 Inverse Hyperbolic Functions
8: 6.4 Analytic Continuation
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►Analytic continuation of the principal value of yields a multi-valued function with branch points at and .
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9: 15.2 Definitions and Analytical Properties
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