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11: 33.13 Complex Variable and Parameters
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►The functions , , and may be extended to noninteger values of by generalizing , and supplementing (33.6.5) by a formula derived from (33.2.8) with expanded via (13.2.42).
►These functions may also be continued analytically to complex values of , , and .
The quantities , , and , given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
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33.13.1
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33.13.2
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12: 3.4 Differentiation
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►If can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))
…where is a simple closed contour described in the positive rotational sense such that and its interior lie in the domain of analyticity of , and is interior to .
Taking to be a circle of radius centered at , we obtain
…The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2).
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►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands.
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13: 26.6 Other Lattice Path Numbers
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Delannoy Number
► is the number of paths from to that are composed of directed line segments of the form , , or . … ►
26.6.12
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26.6.13
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26.6.14
14: DLMF Project News
error generating summary15: 19.36 Methods of Computation
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►Polynomials of still higher degree can be obtained from (19.19.5) and (19.19.7).
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►The cancellations can be eliminated, however, by using (19.25.10).
►Accurate values of for near 0 can be obtained from by (19.2.6) and (19.25.13).
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►The incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6).
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can be evaluated by using (19.25.7), and by using (19.21.10), but cancellations may become significant.
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16: 10.36 Other Differential Equations
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►The quantity in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by if at the same time the symbol in the given solutions is replaced by .
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►Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing by .
17: 23.20 Mathematical Applications
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►An algebraic curve that can be put either into the form
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►Let denote the set of points on that are of finite order (that is, those points for which there exists a positive integer with ), and let
be the sets of points with integer and rational coordinates, respectively.
…Both are subgroups of , though may not be.
…To determine , we make use of the fact that if then must be a divisor of ; hence there are only a finite number of possibilities for .
…The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from , , , , , , or .
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18: 34.9 Graphical Method
§34.9 Graphical Method
… ►Thus, any analytic expression in the theory, for example equations (34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be represented by a diagram; conversely, any diagram represents an analytic equation. …For specific examples of the graphical method of representing sums involving the , and symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).19: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►In what follows will be taken to be a self adjoint extension of following the discussion ending the prior sub-section.
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►These eigenvalues will be assumed distinct, i.
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►this being a matrix element of the resolvent
, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7).
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►Should
be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10).
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►For a formally self-adjoint second order differential operator , such as that of (1.18.28), the space can be seen to consist of all such that the distribution can be identified with a function in , which is the function .
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20: 3.6 Linear Difference Equations
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►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome.
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►Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors.
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►However, can be computed successfully in these circumstances by boundary-value methods, as follows.
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►The least value of that satisfies (3.6.9) is found to be 16.
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►Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability.
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