exceptional values
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11: 2.6 Distributional Methods
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►We may therefore define the integral on the left-hand side of (2.6.4) by the value on the right-hand side, except when .
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12: 1.10 Functions of a Complex Variable
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►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in with at most one exception.
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13: 35.1 Special Notation
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complex variables. |
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determinant of (except when where it means either determinant or absolute value, depending on the context). |
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14: 28.2 Definitions and Basic Properties
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►If , then for a given value of the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)).
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15: 15.2 Definitions and Analytical Properties
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►For all values of
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►Except where indicated otherwise principal branches of and are assumed throughout the DLMF.
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►As a multivalued function of , is analytic everywhere except for possible branch points at , , and .
The same properties hold for , except that as a function of , in general has poles at .
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16: 27.13 Functions
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►Hilbert (1909) proves the existence of for every but does not determine its corresponding numerical value.
The exact value of is now known for every .
…If with , then equality holds in (27.13.2) provided , a condition that is satisfied with at most a finite number of exceptions.
►The existence of follows from that of because , but only the values
and are known exactly.
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►For values of the analysis of is considerably more complicated (see Hardy (1940)).
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17: 16.2 Definition and Analytic Properties
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►When the series (16.2.1) converges for all finite values of and defines an entire function.
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►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at , and .
Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values.
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►In general the series (16.2.1) diverges for all nonzero values of .
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►(However, except where indicated otherwise in the DLMF we assume that when at least one of the is a nonpositive integer.)
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18: 4.37 Inverse Hyperbolic Functions
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►Except where indicated otherwise, it is assumed throughout the DLMF that the inverse hyperbolic functions assume their principal values.
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19: 2.7 Differential Equations
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►See §2.11(v) for other examples.
►The exceptional case is handled by Fabry’s transformation:
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2.7.25
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20: 34.4 Definition: Symbol
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►where the summation is taken over all admissible values of the ’s and ’s for each of the four symbols; compare (34.2.2) and (34.2.3).
►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1.
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