formally self-adjoint operators
(0.002 seconds)
11—20 of 103 matching pages
11: 3.10 Continued Fractions
…
►Every convergent, asymptotic, or formal series
…
►We say that it corresponds to the formal power series
…
►We say that it is associated with the formal power series in (3.10.7) if the expansion of its th convergent in ascending powers of , agrees with (3.10.7) up to and including the term in , .
…
►( is the backward difference operator.)
…
12: 1.17 Integral and Series Representations of the Dirac Delta
…
►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)):
…Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation
…
►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non- improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly.
…
►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)):
…
►By analogy with §1.17(ii) we have the formal series representation
…
13: 16.11 Asymptotic Expansions
…
►
§16.11(i) Formal Series
►For subsequent use we define two formal infinite series, and , as follows: ►
16.11.1
,
►
16.11.2
…
►The formal series (16.11.2) for converges if , and
…
14: Bibliography K
…
►
Lowering and Raising Operators for Some Special Orthogonal Polynomials.
In Jack, Hall-Littlewood and Macdonald Polynomials,
Contemp. Math., Vol. 417, pp. 227–238.
…
►
Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators.
SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
►
Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator.
Adv. Math. 333, pp. 796–821.
…
►
Quantum-Theoretical Formalism for Inhomogeneous Graded-Index Waveguides.
Akademie Verlag, Berlin-New York.
…
►
Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions.
J. Math. Anal. Appl. 324 (1), pp. 285–303.
15: How to Cite
…
►When citing DLMF from a formal publication, we suggest a format similar to the following:
…
16: 18.2 General Orthogonal Polynomials
…
►
§18.2(ii) -Difference Operators
… ►It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. … ►where and are formal power series in , with , and . …If is the formal power series such that then a property equivalent to (18.2.45) with is that … …17: DLMF Project News
error generating summary18: 2.3 Integrals of a Real Variable
…
►In both cases the th error term is bounded in absolute value by , where the variational
operator
is defined by
►
2.3.6
…
►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion:
…
►The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method.
…
19: 2.1 Definitions and Elementary Properties
…
►Let be a formal power series (convergent or divergent) and for each positive integer ,
…
►means that for each , the difference between and the th partial sum on the right-hand side is as in .
►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series.
…
20: 10.70 Zeros
…
►Let and denote the formal series
…