general case
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11: 19.20 Special Cases
12: 4.2 Definitions
13: 15.12 Asymptotic Approximations
14: 32.11 Asymptotic Approximations for Real Variables
15: 20.7 Identities
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►These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183).
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16: Bibliography S
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Relations among the fundamental solutions of the generalized hypergeometric equation when . II. Logarithmic cases.
Bull. Amer. Math. Soc. 45 (12), pp. 927–935.
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Error bounds for asymptotic solutions of differential equations. II. The general case.
J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.
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17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►thus generalizing the inner product of (1.18.9).
When
is absolutely continuous, i.e.
, see §1.4(v), where the nonnegative weight function
is
Lebesgue measurable on
. In this section we will only consider the special case
, so
;
in which case
.
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§1.18(vii) Continuous Spectra: More General Cases
… ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. This work is well overviewed by Coddington and Levinson (1955, Ch. 9), and then applied in detail by Titchmarsh (1946), Titchmarsh (1962a), Titchmarsh (1958), and Levitan and Sargsjan (1975) which also connects the Weyl theory to the relevant functional analysis. In parallel, similar, and more general formulations have grown out of functional analysis itself, as in the work of Stone (1990), Rudin (1973), Reed and Simon (1980), Reed and Simon (1975), Reed and Simon (1978), Reed and Simon (1979), Cycon et al. (2008), Dunford and Schwartz (1988, Ch. XIII), Hall (2013, pp. 127-223). …18: 8.21 Generalized Sine and Cosine Integrals
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►When (and when , in the case of , or , in the case of ) the principal values of , , , and are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)).
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