locally integrable

SearchAdvancedHelp

(0.002 seconds)

31—39 of 39 matching pages

31: 18.39 Applications in the Physical Sciences

… ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. … ►Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. … ►with an infinite set of orthonormal $L^2$ eigenfunctions … ►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. … ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

32: 16.17 Definition

… ►where the integration path $L$ separates the poles of the factors $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ from those of the factors $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$. … ►

16.17.3 $$A_{p,q,k}^{m,n}(z)=\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne k\end{array}}^m\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-b_k\right)\prod _{\mathrm{\ell }=1}^n\mathrm{\Gamma }\left(1+b_k-a_{\mathrm{\ell }}\right)z^{b_k}/\left(\prod _{\mathrm{\ell }=m}^{q-1}\mathrm{\Gamma }\left(1+b_k-b_{\mathrm{\ell }+1}\right)\prod _{\mathrm{\ell }=n}^{p-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }+1}-b_k\right)\right).$$

ⓘ

Defines:

$A_{p,q}^{m,n}(z)$: coefficient (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.17 and Ch.16

33: 6.12 Asymptotic Expansions

… ►

6.12.7 $$R_n^{(\mathrm{f})}(z)={(-1)}^n{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{2n}}{t^2+1}dt,$$

ⓘ

Defines:

$R_n^{(\mathrm{f})}(z)$: remainder term (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

►

6.12.8 $$R_n^{(\mathrm{g})}(z)={(-1)}^n{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{2n+1}}{t^2+1}dt.$$

ⓘ

Defines:

$R_n^{(\mathrm{g})}(z)$: remainder term (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

…

34: 1.4 Calculus of One Variable

… ►

Maxima and Minima

… ►

§1.4(iv) Indefinite Integrals

… ►

Integration by Parts

… ►

§1.4(v) Definite Integrals

… ►

Square-Integrable Functions

…

35: 18.34 Bessel Polynomials

… ►the integration path being taken in the positive rotational sense. …

36: 1.9 Calculus of a Complex Variable

… ►

§1.9(iii) Integration

… ►

1.9.32 $$\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{1}{z-z_0}dz=𝒩(C,z_0),$$

ⓘ

Defines:

$𝒩(C,z_0)$: winding number of $C$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $C$: closed contour

Permalink:

http://dlmf.nist.gov/1.9.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(iii), §1.9(iii), §1.9 and Ch.1

… ►

1.9.49 $$R=\underset{n\to \mathrm{\infty }}{lim\; inf}{\left|a_n\right|}^{-1/n}.$$

ⓘ

Defines:

$R$: radius of convergence (locally)

Symbols:

$lim\; inf$: least limit point, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.9.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(vi), §1.9 and Ch.1

… ►

Term-by-Term Integration

…

37: 7.12 Asymptotic Expansions

… ►

7.12.6 $$R_n^{(\mathrm{f})}(z)=\frac{{(-1)}^n}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}{t}^{2n-(1/2)}}{t^2+1}dt,$$

ⓘ

Defines:

$R_n^{(\mathrm{f})}(z)$: remainder term (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.29 (in different form)

Referenced by:

§7.12(ii)

Permalink:

http://dlmf.nist.gov/7.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

►

7.12.7 $$R_n^{(\mathrm{g})}(z)=\frac{{(-1)}^n}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}{t}^{2n+(1/2)}}{t^2+1}dt.$$

ⓘ

Defines:

$R_n^{(\mathrm{g})}(z)$: remainder term (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.30 (in different form)

Referenced by:

§7.12(ii)

Permalink:

http://dlmf.nist.gov/7.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

…

38: 1.10 Functions of a Complex Variable

… ►and the integration contour is described once in the positive sense. … ►

1.10.13 $$F(w)=z_0+\sum _{n=1}^{\mathrm{\infty }}{F}_n{(w-w_0)}^n$$

ⓘ

Defines:

$F(w)$: inverse function of $f(z)$ (locally)

Symbols:

$z$: variable, $w$: variable, $n$: nonnegative integer and $F_n$: residue

Referenced by:

§1.10(vii)

Permalink:

http://dlmf.nist.gov/1.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(vii), §1.10(vii), §1.10 and Ch.1

… ►is analytic in $D$ and its derivatives of all orders can be found by differentiating under the sign of integration. … ►

1.10.23 $$F(z)=\prod _{n=1}^{\mathrm{\infty }}{a}_n(z),$$ $z\in D$,

ⓘ

Defines:

$F$: function (locally)

Symbols:

$\in$: element of, $z$: variable, $n$: nonnegative integer and $D$: domain

Permalink:

http://dlmf.nist.gov/1.10.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(x), §1.10 and Ch.1

…

39: Bibliography C

… ►

Orthogonalization Procedures and the Localization of Wannier Functions. Phys. Rev. 105, pp. 102–103.

ⓘ

External Links:

Document, Link, ZentralBlatt

Referenced by:

Profile Bille C. Carlson.

Encodings:

BibTeX

… ►

Elliptic Integrals: Symmetry and Symbolic Integration. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti dei Convegni Lincei, Vol. 147, pp. 161–181.

ⓘ

External Links:

MathReview (S. L. Kalla), ZentralBlatt

Referenced by:

§19.26(iii), §19.29(i).

Encodings:

BibTeX

… ►

Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

ⓘ

Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

Encodings:

BibTeX

… ►

A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.

ⓘ

External Links:

ISSN 0029-599X, Document, MathReview (A. H. Stroud), ZentralBlatt

Referenced by:

§3.5(iv).

Encodings:

BibTeX

… ►

On the integration of incomplete elliptic integrals. Proc. Roy. Soc. London Ser. A 444, pp. 525–532.

ⓘ

External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§19.13(i), §19.13(ii).

Encodings:

BibTeX

…