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1: 28.7 Analytic Continuation of Eigenvalues
§28.7 Analytic Continuation of Eigenvalues
… βΊThe branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of are normal values. … βΊ …2: 23.23 Tables
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βΊ2 in Abramowitz and Stegun (1964) gives values of , , and to 7 or 8D in the rectangular and rhombic cases, normalized so that and (rectangular case), or and (rhombic case), for = 1.
…The values are tabulated on the real and imaginary -axes, mostly ranging from 0 to 1 or in steps of length 0.
05, and in the case of the user may deduce values for complex by application of the addition theorem (23.10.1).
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3: William P. Reinhardt
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βΊReinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals.
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βΊIn November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
4: 33.13 Complex Variable and Parameters
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βΊThe 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).
βΊThese functions may also be continued analytically to complex values of , , and .
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βΊ
33.13.1
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βΊ
33.13.2
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5: 28.12 Definitions and Basic Properties
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βΊ
§28.12(i) Eigenvalues
… βΊ … βΊAs in §28.7 values of for which (28.2.16) has simple roots are called normal values with respect to . For real values of and all the are real, and is normal. … βΊIf is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of and by the normalization …6: 8.2 Definitions and Basic Properties
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βΊThe general values of the incomplete gamma functions
and are defined by
…Except where indicated otherwise in the DLMF these principal values are assumed.
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βΊNormalized functions are:
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βΊIn this subsection the functions and have their general values.
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βΊ(8.2.9) also holds when is zero or a negative integer, provided that the right-hand side is replaced by its limiting value.
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7: 3.6 Linear Difference Equations
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βΊIt therefore remains to apply a normalizing factor .
The process is then repeated with a higher value of , and the normalized solutions compared.
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βΊThe normalizing factor can be the true value of divided by its trial value, or can be chosen to satisfy a known property of the wanted solution of the form
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βΊFor further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6).
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8: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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Algorithm 39: Areas under the normal curve.
The Computer Journal 12 (2), pp. 197–198.
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Normal forms of functions near degenerate critical points, the Weyl groups and Lagrangian singularities.
Funkcional. Anal. i PriloΕΎen. 6 (4), pp. 3–25 (Russian).
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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Critical points of smooth functions, and their normal forms.
Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
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9: 28.14 Fourier Series
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βΊand the normalization relation
βΊ
28.14.5
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βΊAmbiguities in sign are resolved by (28.14.9) when , and by continuity for other values of .
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10: 31.9 Orthogonality
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βΊThe branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning.
βΊThe normalization constant is given by
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31.9.3
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βΊThe right-hand side may be evaluated at any convenient value, or limiting value, of in since it is independent of .
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βΊFor further information, including normalization constants, see Sleeman (1966a).
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