formally self adjoint linear operator
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11: 3.10 Continued Fractions
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►Every convergent, asymptotic, or formal series
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►We say that it corresponds to the formal power series
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►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 , .
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►( is the backward difference operator.)
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12: 16.11 Asymptotic Expansions
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§16.11(i) Formal Series
►For subsequent use we define two formal infinite series, and , as follows: ►
16.11.1
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16.11.2
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►The formal series (16.11.2) for converges if , and
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13: 1.17 Integral and Series Representations of the Dirac Delta
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►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
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►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.
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►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)):
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►By analogy with §1.17(ii) we have the formal series representation
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14: How to Cite
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►When citing DLMF from a formal publication, we suggest a format similar to the following:
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15: Bibliography K
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Quasi-linear Stokes phenomenon for the Painlevé first equation.
J. Phys. A 37 (46), pp. 11149–11167.
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Linear convergence and the bisection algorithm.
Amer. Math. Monthly 93 (1), pp. 48–51.
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Lowering and Raising Operators for Some Special Orthogonal Polynomials.
In Jack, Hall-Littlewood and Macdonald Polynomials,
Contemp. Math., Vol. 417, pp. 227–238.
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An algorithm for solving second order linear homogeneous differential equations.
J. Symbolic Comput. 2 (1), pp. 3–43.
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Quantum-Theoretical Formalism for Inhomogeneous Graded-Index Waveguides.
Akademie Verlag, Berlin-New York.
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16: DLMF Project News
error generating summary17: 10.70 Zeros
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►Let and denote the formal series
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18: Bibliography B
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Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations.
The Macmillan Co., New York.
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An Introduction to Linear Difference Equations.
Dover Publications Inc., New York.
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Formal Power Series and Algebraic Combinatorics.
DIMACS Series in Discrete Mathematics and Theoretical Computer
Science, Vol. 24, American Mathematical Society, Providence, RI.
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Discrete ordinate solution of Fokker-Planck equations with non-linear coefficients.
Phys. Rev. A 31 (3), pp. 1855–1868.
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Error estimates for the solution of linear systems.
SIAM J. Sci. Comput. 21 (2), pp. 764–781.
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19: 2.3 Integrals of a Real Variable
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►In both cases the th error term is bounded in absolute value by , where the variational
operator
is defined by
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2.3.6
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►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion:
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►The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method.
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20: 18.2 General Orthogonal Polynomials
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