Dirac%20delta

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21: 1.4 Calculus of One Variable

… ►that is, for every arbitrarily small positive constant $ϵ$ there exists $\delta$ ($>0$) such that …for all $\alpha$ such that . … ►If, for example, $\alpha (x)=H\left(x-x_n\right)$, the Heaviside unit step-function (1.16.14), then the corresponding measure $d\alpha (x)$ is $\delta \left(x-x_n\right)dx$, where $\delta \left(x-x_n\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, ${\int }_a^{b}f(x)d\alpha (x)=f(x_n)$ for $x_n\in (a,b)$ and $0$ otherwise. Delta distributions and Dirac $\delta$-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17. … ►Let $d\alpha (x)=w(x)dx+{\sum }_{n=1}^N{w}_n\delta \left(x-x_n\right)dx$, $x_n\in (a,b)$, $n=1,\mathrm{\dots }N$. …

22: 20.13 Physical Applications

… ►is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at $t=0$. …

23: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ► ►►►►

$ℂ$	complex plane (excluding infinity).
…
${\delta }_{j,k}$ or ${\delta }_{jk}$	Kronecker delta: 0 if $j\ne k$; 1 if $j=k$.
$\mathrm{\Delta }$ (or ${\mathrm{\Delta }}_x$)	forward difference operator: $\mathrm{\Delta }f(x)=f(x+1)-f(x)$.
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24: 25.21 Software

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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals

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25: 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.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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26: 18.39 Applications in the Physical Sciences

… ►These eigenfunctions are quantum wave-functions whose absolute values squared give the probability density of finding the single particle at hand at position $x$ in the $n$th eigenstate, namely that probability is $P(x\to x+\mathrm{\Delta }x)$ = ${|{\psi }_n(x)|}^2\mathrm{\Delta }(x)$, $\mathrm{\Delta }(x)$ being a localized interval on the $x$-axis. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. … ►With $N\to \mathrm{\infty }$ the functions normalized as $\delta (ϵ-{ϵ}^{\prime })$ with measure $dr$ are, formally, …

27: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►

18.36.6 $${\int }_0^{\mathrm{\infty }}{\widehat{L}}_n^{(k)}\left(x\right){\widehat{L}}_m^{(k)}\left(x\right){\widehat{W}}_k(x)dx=\frac{(n+k)\mathrm{\Gamma }\left(n+k-1\right)}{(n-1)!}{\delta }_{n,m}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , ${\widehat{L}}_n^{(k)}\left(x\right)$: exceptional Laguerre polynomial, $!$: factorial (as in $n!$), $\int$: integral, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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28: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Algorithm 779: Fermi-Dirac functions of order $-1/2$, $1/2$, $3/2$, $5/2$ . ACM Trans. Math. Software 24 (1), pp. 1–12.

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Notes:

Double-precision Fortran, maximum accuracy 16D.

External Links:

ISSN 0098-3500, Document, GAMS (module 779; package TOMS), ZentralBlatt

Referenced by:

8th item.

Encodings:

BibTeX

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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The accurate evaluation of a particular Fermi-Dirac integral. Comput. Phys. Comm. 101 (1-2), pp. 47–53.

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External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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29: Bibliography S

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A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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On the evaluation of the Fermi-Dirac integrals. Astrophys. J. 376 (1, part 1), pp. 364–366.

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External Links:

ISSN 0004-637X, Document, MathReview (De Hui Chen)

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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30: Software Index

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	Open Source													With Book						Commercial
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20 Theta Functions
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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
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