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§1.17 Integral and Series Representations of the Dirac Delta

►

§1.17(i) Delta Sequences

… ►for a suitably chosen sequence of functions

\delta_{n}\left(x\right)

,

n=1,2,\dots

. Such a sequence is called a delta sequence and we write, symbolically, … ►An example of a delta sequence is provided by …

… ►

§3.9(i) Sequence Transformations

►All sequences (series) in this section are sequences (series) of real or complex numbers. … ►

§3.9(iii) Aitken’s $\Delta^{2}$ -Process

… ►, a sequence for which … ►Sequences that are accelerated by Levin’s transformation include logarithmically convergent sequences, i. …

… ►for all

n

sufficiently large, where

A

and

p

are independent of

n

, then the sequence is said to have convergence of the

p

th order. … ►For real functions

f(x)

the sequence of approximations to a real zero

\xi

will always converge (and converge quadratically) if either: … ►Then the next approximation to the quadratic factor is

z^{2}-(s+\Delta s)z-(t+\Delta t)

, where … ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. …

… ►Many special functions satisfy second-order recurrence relations, or difference equations, of the form … ►

3.6.2

a_{n}\Delta^{2}w_{n-1}+(2a_{n}-b_{n})\Delta w_{n-1}+(a_{n}-b_{n}+c_{n})w_{n-1}% =d_{n},

ⓘ

Symbols:: $\Delta$ : forward difference operator, $w_{n}$ : sequence, $a_{n}$ : coefficient, $b_{n}$ : coefficient, $c_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §3.6(i), §3.6 and Ch.3

►where

\Delta w_{n-1}=w_{n}-w_{n-1}

,

\Delta^{2}w_{n-1}=\Delta w_{n}-\Delta w_{n-1}

, and

n\in\mathbb{Z}

. … ►At the same time we construct a sequence

e_{n}

,

n=0,1,\dots

, defined by … ►

3.6.10

p_{n+1}w_{n}=p_{n}w_{n+1}+e_{n},

ⓘ

Symbols:: $w_{n}$ : sequence, $p_{n}$ : solutions and $e_{n}$ : sequence
Permalink:: http://dlmf.nist.gov/3.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §3.6(v), §3.6 and Ch.3

…

… ►

b_{0}+\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

is equivalent to

b^{\prime}_{0}+\displaystyle{\cfrac{a^{\prime}_{1}}{b^{\prime}_{1}+\cfrac{a^{% \prime}_{2}}{b^{\prime}_{2}+\cdots}}}

if there is a sequence

\{d_{n}\}^{\infty}_{n=0}

,

d_{0}=1

,

d_{n}\neq 0

, such that … ►A sequence

\{C_{n}\}

in the extended complex plane,

\mathbb{C}\cup\{\infty\}

, can be a sequence of convergents of the continued fraction (1.12.3) iff … ►

1.12.26

-\tfrac{1}{2}\pi+\delta<\operatorname{ph}b_{n}<\tfrac{1}{2}\pi-\delta,

n=1,2,3,\dots

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

►where

\delta

is an arbitrary small positive constant. … ►

1.12.27

-\tfrac{1}{2}\pi+\delta<\operatorname{ph}C_{n}<\tfrac{1}{2}\pi-\delta,

n=1,2,3,\dots

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $n$ : nonnegative integer and $C_{n}$ : approximant
Permalink:: http://dlmf.nist.gov/1.12.E27
Encodings:: pMML, png, TeX
See also:: Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

…

… ►

M. N. Olevskiĭ (1950) Triorthogonal systems in spaces of constant curvature in which the equation

\Delta_{2}u+\lambda u=0

allows a complete separation of variables. Mat. Sbornik N.S. 27(69) (3), pp. 379–426 (Russian).

ⓘ

Notes:: In Russian.
External Links:: MathReview (H. Busemann), ZentralBlatt
Referenced by:: §31.17(i).
Encodings:: BibTeX

… ►

J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

ⓘ

External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

… ►

On-Line Encyclopedia of Integer Sequences (website) OEIS Foundation, Inc., Highland Park, New Jersey.

ⓘ

Notes:: A tool for finding and identifying integer sequences.
External Links:: On-Line Encyclopedia of Integer Sequences
Referenced by:: 5th item.
Encodings:: BibTeX

…

… ►We say that a sequence of distributions

\{\Lambda_{n}\}

converges to a distribution

\Lambda

in

\mathcal{D}^{*}

if … ►

§1.16(iii) Dirac Delta Distribution

… ►The Dirac delta distribution is singular. … ►We use the notation of the previous subsection and take

n=1

and

u=\delta

in (1.16.35). … ►Since

\sqrt{2\pi}\mathscr{F}\left(\delta\right)=1

, we have …

… ►

G. A. Kalugin and D. J. Jeffrey (2011) Unimodal sequences show that Lambert

W

is Bernstein. C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.

ⓘ

External Links:: ISSN 0706-1994, MathReview (Fana Tangara), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

T. H. Koornwinder (1984b) Orthogonal polynomials with weight function

(1-x)^{\alpha}(1+x)^{\beta}+M\delta(x+1)+N\delta(x-1)

. Canad. Math. Bull. 27 (2), pp. 205–214.

ⓘ

External Links:: ISSN 0008-4395, MathReview (Wolfgang Hahn), ZentralBlatt
Referenced by:: §18.36(i), §18.36(i).
Encodings:: BibTeX

…

… ►

R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

… ►

W. J. Cody (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 24S.
External Links:: ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

Combinatorial Object Server (website) Department of Computer Science, University of Victoria, Canada.

ⓘ

Notes:: Permutations, set and integer partitions, sequences, graphs, and trees.
External Links:: Combinatorial Object Server
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

R. M. Corless, D. J. Jeffrey, and D. E. Knuth (1997) A sequence of series for the Lambert

W

function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

ⓘ

External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.
Encodings:: BibTeX

…

… ►In (2.1.5)

\mathbb{R}

can be replaced by any fixed ray in the sector

|\operatorname{ph}x|<\frac{1}{2}\pi

, or by the whole of the sector

|\operatorname{ph}x|\leq\frac{1}{2}\pi-\delta

. (Here and elsewhere in this chapter

\delta

is an arbitrary small positive constant.) … ►Let

\phi_{s}(x)

,

s=0,1,2,\dots

, be a sequence of functions defined in

\mathbf{X}

such that for each

s

…Then

\{\phi_{s}(x)\}

is an asymptotic sequence or scale. Suppose also that

f(x)

and

f_{s}(x)

satisfy …

delta%20sequences

1—10 of 299 matching pages

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

2: 3.9 Acceleration of Convergence

§3.9(i) Sequence Transformations

§3.9(iii) Aitken’s $\Delta^{2}$ -Process

3: 3.8 Nonlinear Equations

4: 3.6 Linear Difference Equations

5: 1.12 Continued Fractions

6: Bibliography O

7: 1.16 Distributions

§1.16(iii) Dirac Delta Distribution

8: Bibliography K

9: Bibliography C

10: 2.1 Definitions and Elementary Properties

delta%20sequences

1—10 of 299 matching pages

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

2: 3.9 Acceleration of Convergence

§3.9(i) Sequence Transformations

§3.9(iii) Aitken’s Δ 2 -Process

3: 3.8 Nonlinear Equations

4: 3.6 Linear Difference Equations

5: 1.12 Continued Fractions

6: Bibliography O

7: 1.16 Distributions

§1.16(iii) Dirac Delta Distribution

8: Bibliography K

9: Bibliography C

10: 2.1 Definitions and Elementary Properties

§3.9(iii) Aitken’s $\Delta^{2}$ -Process