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1: 1.17 Integral and Series Representations of the Dirac Delta
§1.17 Integral and Series Representations of the Dirac Delta
►§1.17(i) Delta Sequences
… ►for a suitably chosen sequence of functions , . Such a sequence is called a delta sequence and we write, symbolically, … ►An example of a delta sequence is provided by …2: 3.9 Acceleration of Convergence
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§3.9(i) Sequence Transformations
►All sequences (series) in this section are sequences (series) of real or complex numbers. … ►§3.9(iii) Aitken’s -Process
… ►, a sequence for which … ►Sequences that are accelerated by Levin’s transformation include logarithmically convergent sequences, i. …3: 3.8 Nonlinear Equations
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►for all sufficiently large, where and are independent of , then the sequence is said to have convergence of the
th order.
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►For real functions the sequence of approximations to a real zero will always converge (and converge quadratically) if either:
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►Then the next approximation to the quadratic factor is , where
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►Consider and .
We have and .
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4: 3.6 Linear Difference Equations
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►Many special functions satisfy second-order recurrence relations, or difference equations, of the form
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3.6.2
►where , , and .
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►At the same time we construct a sequence
, , defined by
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3.6.10
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5: 1.12 Continued Fractions
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is equivalent to if there is a sequence
, ,
, such that … ►A sequence in the extended complex plane, , can be a sequence of convergents of the continued fraction (1.12.3) iff … ►
, such that … ►A sequence in the extended complex plane, , can be a sequence of convergents of the continued fraction (1.12.3) iff … ►
1.12.26
,
►where is an arbitrary small positive constant.
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1.12.27
,
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6: Bibliography O
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Triorthogonal systems in spaces of constant curvature in which the equation allows a complete separation of variables.
Mat. Sbornik N.S. 27(69) (3), pp. 379–426 (Russian).
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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OEIS Foundation, Inc., Highland Park, New Jersey.
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7: 1.16 Distributions
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►We say that a sequence of distributions
converges to a distribution in if
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§1.16(iii) Dirac Delta Distribution
… ►The Dirac delta distribution is singular. … ►We use the notation of the previous subsection and take and in (1.16.35). … ►Since , we have …8: Bibliography K
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Unimodal sequences show that Lambert is Bernstein.
C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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Orthogonal polynomials with weight function
.
Canad. Math. Bull. 27 (2), pp. 205–214.
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9: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Algorithm 597: Sequence of modified Bessel functions of the first kind.
ACM Trans. Math. Software 9 (2), pp. 242–245.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Department of Computer Science, University of Victoria, Canada.
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A sequence of series for the Lambert function.
In Proceedings of the 1997 International Symposium on
Symbolic and Algebraic Computation (Kihei, HI),
pp. 197–204.
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10: 2.1 Definitions and Elementary Properties
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►In (2.1.5) can be replaced by any fixed ray in the sector , or by the whole of the sector .
(Here and elsewhere in this chapter is an arbitrary small positive constant.)
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►Let , , be a sequence of functions defined in such that for each
…Then is an asymptotic sequence or scale.
Suppose also that and satisfy
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