completness relation

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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

\delta\left(x-x^{\prime}\right)=\sum_{\boldsymbol{\sigma}_{p}}\phi_{\lambda_{n% }}(x)\overline{\phi_{\lambda_{n}}(x^{\prime})}+\int_{\boldsymbol{\sigma}_{c}}% \phi_{\lambda}(x)\overline{\phi_{\lambda}(x^{\prime})}\,\mathrm{d}\lambda,

x,x^{\prime}\in X

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.18.E63
Encodings:: pMML, png, TeX
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

…

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

\zeta=K

K+i{K^{\prime}}

, and

i{K^{\prime}}

, where

K

and

{K^{\prime}}

are related to

k

as in §19.2(ii).

4: Bibliography B

… ►

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.

ⓘ

External Links:: ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.13(i).
Encodings:: BibTeX

…

5: 22.8 Addition Theorems

… ►If sums/differences of the

z_{j}

’s are rational multiples of

K\left(k\right)

, then further relations follow. …

6: 22.16 Related Functions

… ►

22.16.28

\mathcal{E}\left(x+K,k\right)=\mathcal{E}\left(x,k\right)+E\left(k\right)-k^{2% }\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right),

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

…

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

… ►

§14.5(v) $\mu=0$ , $\nu=\pm\frac{1}{2}$

►In this subsection

K\left(k\right)

and

E\left(k\right)

denote the complete elliptic integrals of the first and second kinds; see §19.2(ii). ►

14.5.20

\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}\left(2E\left(% \sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(\sin\left(\tfrac{1}{2}\theta% \right)\right)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function and $0<\theta<\pi$ : variable
A&S Ref:: 8.13.11 (in slightly different form)
Permalink:: http://dlmf.nist.gov/14.5.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §14.5(v), §14.5 and Ch.14

… ►

14.5.22

\mathsf{Q}_{\frac{1}{2}}\left(\cos\theta\right)=K\left(\cos\left(\tfrac{1}{2}% \theta\right)\right)-2E\left(\cos\left(\tfrac{1}{2}\theta\right)\right),

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind and $0<\theta<\pi$ : variable
A&S Ref:: 8.13.12 (in slightly different form)
Permalink:: http://dlmf.nist.gov/14.5.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §14.5(v), §14.5 and Ch.14

…

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). …