高中毕业证等级h《做证微fuk7778》CZ0F
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11: 10.2 Definitions
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►These solutions of (10.2.1) are denoted by and , and their defining properties are given by
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►The principal branches of and are two-valued and discontinuous on the cut .
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►For fixed
each branch of and is entire in .
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►Except where indicated otherwise, it is assumed throughout the DLMF that the symbols , , , and denote the principal values of these functions.
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►The notation denotes , , , , or any nontrivial linear combination of these functions, the coefficients in which are independent of and .
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12: 28.26 Asymptotic Approximations for Large
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28.26.1
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28.26.2
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►Then as with fixed in and fixed ,
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►The asymptotic expansions of and in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively.
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►For asymptotic approximations for see also Naylor (1984, 1987, 1989).
13: 28.1 Special Notation
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►The functions and are also known as the radial Mathieu functions.
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►The radial functions and are denoted by and , respectively.
integers. | |
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real or complex parameters of Mathieu’s equation with . | |
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14: 29.9 Stability
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►The Lamé equation (29.2.1) with specified values of is called stable if all of its solutions are bounded on ; otherwise the equation is called unstable.
If is not an integer, then (29.2.1) is unstable iff or lies in one of the closed intervals with endpoints and , .
If is a nonnegative integer, then (29.2.1) is unstable iff or for some .
15: 28.25 Asymptotic Expansions for Large
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►For fixed and fixed ,
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28.25.3
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►The upper signs correspond to and the lower signs to .
The expansion (28.25.1) is valid for when
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16: 28.28 Integrals, Integral Representations, and Integral Equations
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28.28.2
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►In particular, when the integrals (28.28.11), (28.28.14) converge absolutely and uniformly in the half strip , .
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28.28.16
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►With the parameter suppressed we use the notation
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►Again with the parameter suppressed, let
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17: 10.47 Definitions and Basic Properties
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and are the spherical Bessel
functions of the first and second kinds, respectively; and are the spherical Bessel functions of the
third kind.
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►Many properties of , , , , , , and follow straightforwardly from the above definitions and results given in preceding sections of this chapter.
For example, , , , , , , and are all entire functions of .
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►For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols , , , and replaced by , , , and , respectively.
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10.47.15
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18: 10.52 Limiting Forms
19: 28.10 Integral Equations
20: 28.20 Definitions and Basic Properties
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28.20.8
►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant.
It follows that (28.20.1) has independent and unique solutions , such that
…In addition, there are unique solutions , that are real when is real and have the properties
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►For other values of , , and the functions , , are determined by analytic continuation.
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