along the real line
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11—18 of 18 matching pages
11: 4.15 Graphics
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§4.15(i) Real Arguments
… ►Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole -plane cut along the real axis from to and to , where and (principal value). …Lines parallel to the real axis in the -plane map onto ellipses in the -plane with foci at , and lines parallel to the imaginary axis in the -plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points is a real parameter that can lie anywhere in the interval . … ► …12: 36.11 Leading-Order Asymptotics
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►With real critical points (36.4.1) ordered so that
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Asymptotics along Symmetry Lines
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36.11.5
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36.11.7
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36.11.8
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13: 31.11 Expansions in Series of Hypergeometric Functions
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31.11.3
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31.11.7
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31.11.12
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►Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function.
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►The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities and .
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14: 36.5 Stokes Sets
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►where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise.
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►The second sheet corresponds to and it intersects the bifurcation set (§36.4) smoothly along the line generated by , .
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►the intersection lines with the bifurcation set are generated by , .
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►The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions.
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15: 10.20 Uniform Asymptotic Expansions for Large Order
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§10.20(i) Real Variables
… ►As through positive real values … ►The function given by (10.20.2) and (10.20.3) can be continued analytically to the -plane cut along the negative real axis. … ►The curves and in the -plane are the inverse maps of the line segments … ►As through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for , the coefficients , , , and , being the analytic continuations of the functions defined in §10.20(i) when is real. …16: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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►The focal line is given by , , and the rays , are given by , .
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30.13.3
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30.13.11
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►Moreover, has to be bounded along the -axis away from the focal line: this requires to be bounded when .
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►If , then this property holds outside the focal line.
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17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►The space is now the full real line, .
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►This will be generalized, along with the choice of , in §1.18(vii).
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►For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function.
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►Suppose that is the whole real line in one dimension, and that , in (1.18.28) has (non-oscillatory) limits of at both , and thus a continuous spectrum on .
…Surprisingly, if on any interval on the real line, even if positive elsewhere, as long as , see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding eigenfunction.
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