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1: 28.7 Analytic Continuation of Eigenvalues
The 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 q are normal values. …
2: 28.19 Expansions in Series of me ν + 2 n Functions
Let q be a normal value28.12(i)) with respect to ν , and f ( z ) be a function that is analytic on a doubly-infinite open strip S that contains the real axis. …
3: 28.12 Definitions and Basic Properties
28.12.2 λ ν ( - q ) = λ ν ( q ) = λ - ν ( q ) .
As in §28.7 values of q for which (28.2.16) has simple roots λ are called normal values with respect to ν . For real values of ν and q all the λ ν ( q ) are real, and q is normal. … If q is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of z and q by the normalization
4: 28.11 Expansions in Series of Mathieu Functions
Let f ( z ) be a 2 π -periodic function that is analytic in an open doubly-infinite strip S that contains the real axis, and q be a normal value28.7). …
5: 19.31 Probability Distributions
R G ( x , y , z ) and R F ( x , y , z ) occur as the expectation values, relative to a normal probability distribution in 2 or 3 , of the square root or reciprocal square root of a quadratic form. …
6: 8.2 Definitions and Basic Properties
The general values of the incomplete gamma functions γ ( a , z ) and Γ ( a , z ) are defined by …Except where indicated otherwise in the DLMF these principal values are assumed. … Normalized functions are: … In this subsection the functions γ and Γ have their general values. … (8.2.9) also holds when a is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. …
7: 3.6 Linear Difference Equations
The process is then repeated with a higher value of N , and the normalized solutions compared. … The normalizing factor Λ can be the true value of w 0 divided by its trial value, or Λ can be chosen to satisfy a known property of the wanted solution of the form …
8: 31.9 Orthogonality
The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. The normalization constant θ m is given by
31.9.3 θ m = ( 1 - e 2 π i γ ) ( 1 - e 2 π i δ ) ζ γ ( 1 - ζ ) δ ( ζ - a ) ϵ f 0 ( q , ζ ) f 1 ( q , ζ ) q 𝒲 { f 0 ( q , ζ ) , f 1 ( q , ζ ) } | q = q m ,
The right-hand side may be evaluated at any convenient value, or limiting value, of ζ in ( 0 , 1 ) since it is independent of ζ . … For further information, including normalization constants, see Sleeman (1966a). …
9: 1.6 Vectors and Vector-Valued Functions
§1.6 Vectors and Vector-Valued Functions
§1.6(iii) Vector-Valued Functions
An orientable surface is oriented if suitable normals have been chosen. …
Stokes’s Theorem
10: 28.2 Definitions and Basic Properties
For nonnegative real values of q , see Figure 28.2.1. …
Distribution
Change of Sign of q
For simple roots q of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations … …