Dirac%20delta%20distribution

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11: 2.6 Distributional Methods

§2.6 Distributional Methods

… ►The Dirac delta distribution in (2.6.17) is given by … ►Furthermore, $K^{+}$ contains the distributions $H$, $\delta$, and $t^{\lambda }$, $t>0$, for any real (or complex) number $\lambda$, where $H$ is the distribution associated with the Heaviside function $H\left(t\right)$ (§1.16(iv)), and $t^{\lambda }$ is the distribution defined by (2.6.12)–(2.6.14), depending on the value of $\lambda$. Since $\delta =𝐷H$, it follows that for $\mu \ne 1,2,\mathrm{\dots }$, … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►of the Dirac delta distribution. … ►Applying the representation (1.17.13), now symmetrized as in (1.17.14), as $\frac{1}{x}\delta \left(x-y\right)=\frac{1}{\sqrt{xy}}\delta \left(x-y\right)$, … ►These latter results also correspond to use of the $\delta \left(x-y\right)$ as defined in (1.17.12_1) and (1.17.12_2). … ►Thus, and this is a case where $q(x)$ is not continuous, if $q(x)=-\alpha \delta \left(x-a\right)$, $\alpha >0$, there will be an $L^2$ eigenfunction localized in the vicinity of $x=a$, with a negative eigenvalue, thus disjoint from the continuous spectrum on $[0,\mathrm{\infty })$. … ►For fixed angular momentum $\mathrm{\ell }$ the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues ${\lambda }_n,n=0,1,\mathrm{\dots },N-1$, with corresponding $L^2([0,\mathrm{\infty }),r^{2}dr)$ eigenfunctions ${\varphi }_n(r)$, and also a continuous spectrum $\lambda \in [0,\mathrm{\infty })$, with Dirac-delta normalized eigenfunctions ${\varphi }_{\lambda }(r)$, also with measure $r^{2}dr$. …

13: Bibliography G

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On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

Encodings:

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

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Algorithm 745: Computation of the complete and incomplete Fermi-Dirac integral. ACM Trans. Math. Software 21 (3), pp. 221–232.

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Notes:

Double-precision Fortran, maximum accuracy 20D.

External Links:

ISSN 0098-3500, Document, GAMS (module 745; package TOMS), ZentralBlatt

Referenced by:

6th item.

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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

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Notes:

Double-precision Fortran, maximum accuracy 20D

External Links:

Document, Code, ZentralBlatt

Referenced by:

7th item.

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

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14: 33.1 Special Notation

… ► ►►►

$k,\mathrm{\ell }$	nonnegative integers.
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$\delta \left(x\right)$	Dirac delta; see §1.17.
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15: Bibliography F

… ►

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

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Generalized Fermi-Dirac integrals—FD, FDG, FDH. Comput. Phys. Comm. 39 (2), pp. 181–185.

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External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

4th item.

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16: Bibliography N

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On the numerical evaluation of the generalised Fermi-Dirac integrals. Comput. Phys. Comm. 76 (1), pp. 48–50.

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External Links:

ISSN 0010-4655, Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

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17: 8 Incomplete Gamma and Related
Functions

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18: 28 Mathieu Functions and Hill’s Equation

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19: Bibliography D

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

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The Fermi-Dirac integrals ${ℱ}_p(\eta )={(p!)}^{-1}{\int }_0^{\mathrm{\infty }}{ϵ}^p{(e^{ϵ-\eta }+1)}^{-1}𝑑ϵ$ . Appl. Sci. Res. B. 6, pp. 225–239.

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External Links:

ISSN 0003-6994, MathReview Entry, ZentralBlatt

Referenced by:

(25.12.14), §25.12(iii), §25.12.

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

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20: 25.19 Tables

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Morris (1979) tabulates ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $\pm x=0.02(.02)1(.1)6$ to 30D.

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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