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11: 2.6 Distributional Methods
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 α , β = 0 , 1 , 2 , . …
12: 1.10 Functions of a Complex Variable
In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in with at most one exception. …
13: 35.1 Special Notation
a , b

complex variables.

| 𝐗 |

determinant of 𝐗 (except when m = 1 where it means either determinant or absolute value, depending on the context).

14: 28.2 Definitions and Basic Properties
If q 0 , then for a given value of ν the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …
15: 15.2 Definitions and Analytical Properties
For all values of c Except where indicated otherwise principal branches of F ( a , b ; c ; z ) and 𝐅 ( a , b ; c ; z ) are assumed throughout the DLMF. … As a multivalued function of z , 𝐅 ( a , b ; c ; z ) is analytic everywhere except for possible branch points at z = 0 , 1 , and . The same properties hold for F ( a , b ; c ; z ) , except that as a function of c , F ( a , b ; c ; z ) in general has poles at c = 0 , 1 , 2 , . …
16: 27.13 Functions
Hilbert (1909) proves the existence of g ( k ) for every k but does not determine its corresponding numerical value. The exact value of g ( k ) is now known for every k 200 , 000 . …If 3 k = q 2 k + r with 0 < r < 2 k , then equality holds in (27.13.2) provided r + q 2 k , a condition that is satisfied with at most a finite number of exceptions. The existence of G ( k ) follows from that of g ( k ) because G ( k ) g ( k ) , but only the values G ( 2 ) = 4 and G ( 4 ) = 16 are known exactly. … For values of k > 24 the analysis of r k ( n ) is considerably more complicated (see Hardy (1940)). …
17: 16.2 Definition and Analytic Properties
When p q the series (16.2.1) converges for all finite values of z and defines an entire function. … Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at z = 0 , 1 , and . Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … In general the series (16.2.1) diverges for all nonzero values of z . … (However, except where indicated otherwise in the DLMF we assume that when p > q + 1 at least one of the a k is a nonpositive integer.) …
18: 4.37 Inverse Hyperbolic Functions
Except where indicated otherwise, it is assumed throughout the DLMF that the inverse hyperbolic functions assume their principal values. …
19: 2.7 Differential Equations
See §2.11(v) for other examples. The exceptional case f 0 2 = 4 g 0 is handled by Fabry’s transformation: …
2.7.25 𝒱 a j , x ( F ) = | a j x | 1 f 1 / 4 ( t ) d 2 d t 2 ( 1 f 1 / 4 ( t ) ) g ( t ) f 1 / 2 ( t ) | d t | .
20: 34.4 Definition: 6 j Symbol
where the summation is taken over all admissible values of the m ’s and m ’s for each of the four 3 j symbols; compare (34.2.2) and (34.2.3). Except in degenerate cases the combination of the triangle inequalities for the four 3 j symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths j 1 , j 2 , j 3 , l 1 , l 2 , l 3 ; see Figure 34.4.1. …