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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
►Equation (1.18.19) is often called the completeness relation. The analogous orthonormality is … ►and completeness relation►The formal completeness relation is now ►
1.18.63 δ ⁡ ( x x ) = 𝝈 p ϕ λ n ⁢ ( x ) ⁢ ϕ λ n ⁢ ( x ) ¯ + 𝝈 c ϕ λ ⁢ ( x ) ⁢ ϕ λ ⁢ ( x ) ¯ ⁢ d λ , x , x X ,
2: 1.17 Integral and Series Representations of the Dirac Delta
►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …
3: 31.7 Relations to Other Functions
►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities ζ = K ⁡ , K ⁡ + i ⁢ K ⁡ , and i ⁢ K ⁡ , where K ⁡ and K ⁡ are related to k as in §19.2(ii).
4: Bibliography B
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  • P. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.
  • 5: 22.8 Addition Theorems
    ►If sums/differences of the z j ’s are rational multiples of K ⁡ ( k ) , then further relations follow. …
    6: 22.16 Related Functions
    7: Wolter Groenevelt
    ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …
    8: 14.5 Special Values
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    §14.5(v) μ = 0 , ν = ± 1 2
    ►In this subsection K ⁡ ( k ) and E ⁡ ( k ) denote the complete elliptic integrals of the first and second kinds; see §19.2(ii). ► ►
    14.5.22 𝖰 1 2 ⁡ ( cos ⁡ θ ) = K ⁡ ( cos ⁡ ( 1 2 ⁢ θ ) ) 2 ⁢ E ⁡ ( cos ⁡ ( 1 2 ⁢ θ ) ) ,
    9: Gergő Nemes
    ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …
    10: 13.8 Asymptotic Approximations for Large Parameters
    ►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). …