normal values
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1: 28.7 Analytic Continuation of Eigenvalues
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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.
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2: 28.19 Expansions in Series of Functions
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βΊLet be a normal value (§28.12(i)) with respect to , and be a function that is analytic on a doubly-infinite open strip that contains the real axis.
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3: 28.12 Definitions and Basic Properties
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28.12.2
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.
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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
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4: 28.11 Expansions in Series of Mathieu Functions
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βΊLet be a -periodic function that is analytic in an open doubly-infinite strip that contains the real axis, and be a normal value (§28.7).
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5: 19.31 Probability Distributions
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and occur as the expectation values, relative to a normal probability distribution in or , of the square root or reciprocal square root of a quadratic form.
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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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βΊThe 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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8: 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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