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11: 19.20 Special Cases
The general lemniscatic case is … The general lemniscatic case is …
12: 4.2 Definitions
where ph z [ π , π ] for the principal value of z a , and is unrestricted in the general case. …
13: 15.12 Asymptotic Approximations
For the more general case in which a 2 = o ( c ) and b 2 = o ( c ) see Wagner (1990). …
14: 32.11 Asymptotic Approximations for Real Variables
In the generic case
15: 20.7 Identities
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). …
16: Bibliography S
  • F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when p = q + 1 . II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.
  • F. Stenger (1966b) Error bounds for asymptotic solutions of differential equations. II. The general case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.
  • 17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    thus generalizing the inner product of (1.18.9). When α is absolutely continuous, i.e. d α ( x ) = w ( x ) d x , see §1.4(v), where the nonnegative weight function w ( x ) is Lebesgue measurable on X . In this section we will only consider the special case w ( x ) = 1 , so d α ( x ) = d x ; in which case L 2 ( X ) L 2 ( X , d x ) .
    §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
    When ph z = 0 (and when a 1 , 3 , 5 , , in the case of Si ( a , z ) , or a 0 , 2 , 4 , , in the case of Ci ( a , z ) ) the principal values of si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). …
    19: 25.11 Hurwitz Zeta Function
    For the more general case ζ ( m , a ) , m = 1 , 2 , , see Elizalde (1986). …
    20: 14.29 Generalizations
    §14.29 Generalizations
    14.29.1 ( 1 z 2 ) d 2 w d z 2 2 z d w d z + ( ν ( ν + 1 ) μ 1 2 2 ( 1 z ) μ 2 2 2 ( 1 + z ) ) w = 0
    are called Generalized Associated Legendre Functions. As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when μ 1 = μ 2 = μ . … …