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11: 17.1 Special Notation

… ►The main functions treated in this chapter are the basic hypergeometric (or

q

-hypergeometric) function

{{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, the bilateral basic hypergeometric (or bilateral

q

-hypergeometric) function

{{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)

, and the

q

-analogs of the Appell functions

\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)

\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)

\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)

, and

\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)

. … ►Another function notation used is the “idem” function: ►

f(\chi_{1};\chi_{2},\dots,\chi_{n})+\operatorname{idem}\left(\chi_{1};\chi_{2}% ,\dots,\chi_{n}\right)=\sum_{j=1}^{n}f(\chi_{j};\chi_{1},\chi_{2},\dots,\chi_{% j-1},\chi_{j+1},\dots,\chi_{n}).

… ►Fine (1988) uses

F(a,b;t:q)

for a particular specialization of a

{{}_{2}\phi_{1}}

function.

12: 18.39 Applications in the Physical Sciences

… ►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►This seems odd at first glance as

\hat{H}_{n+3}\left(x\right)

is a polynomial of order

n+3

for

n=0,1,2,\ldots

, seemingly suggesting that for

n=0

, this being the first excited state, i. … ►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. … ►A relativistic treatment becoming necessary as

Z

becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order

\left(\alpha Z\right)^{2}\sim\left(Z/137\right)^{2}

\alpha

being the dimensionless fine structure constant

{e}^{2}/(4\pi\varepsilon_{0}\hbar c)

, where

c

is the speed of light. … ►This operator may be discretized by projecting it onto the sub-space defined by the first

N

members,

n=0,1,2,\dots,N-1

, of the complete basis of (18.39.44), the eigenfunctions, may be expressed as …

13: 19.36 Methods of Computation

… ►

§19.36(ii) Quadratic Transformations

… ►Thompson (1997, pp. 499, 504) uses descending Landen transformations for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

. … ►The function

\operatorname{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). … ►Bulirsch (1969a, b) extend Bartky’s transformation to

\operatorname{el3}\left(x,k_{c},p\right)

by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

14: 17.6 ${{}_{2}\phi_{1}}$ Function

… ►

Heine’s Third Transformation

… ►

Fine’s First Transformation

… ►

Fine’s Second Transformation

… ►

Fine’s Third Transformation

… ►

Rogers–Fine Identity

…

15: 8 Incomplete Gamma and Related
Functions

…

16: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

… ►

•

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

►

•

Kerimov and Skorokhodov (1984c) tabulates all zeros of $I_{-n-\frac{1}{2}}\left(z\right)$ and $I_{-n-\frac{1}{2}}'\left(z\right)$ in the sector $0\leq\operatorname{ph}z\leq\tfrac{1}{2}\pi$ for $n=1(1)20$ , 9S.

… ►

•

Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , 8D.

17: 8.26 Tables

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

►

•

Chiccoli et al. (1988) presents a short table of $E_{p}\left(x\right)$ for $p=-\tfrac{9}{2}(1)-\tfrac{1}{2}$ , $0\leq x\leq 200$ to 14S.

►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

18: Guide to Searching the DLMF

… ►For example, the expression

{\operatorname{Ai}}^{2}+{\operatorname{Bi}}^{2}

does not occur verbatim in DLMF, but

{\operatorname{Ai}}^{2}\left(z\right)+{\operatorname{Bi}}^{2}\left(z\right)

and

{\operatorname{Ai}}^{2}\left(x\right)+{\operatorname{Bi}}^{2}\left(x\right)

do. Therefore, if your query is Ai^2+Bi^2, the system modifies the query so it will find the equations containing the latter expressions. ►

Fine Points in Math Search

… ►For example, for the Bessel function

K_{n}\left(z\right)

, you can write K_n(z), BesselK_n(z), BesselK(n,z), or BesselK[n,z]. Note that the first form may match other functions

K

than the Bessel

K

function, so if you are sure you want Bessel

K

, you might as well enter one of the other 3 forms. …

19: 20.10 Integrals

… ►

§20.10(i) Mellin Transforms with respect to the Lattice Parameter

►

20.10.1

\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E1
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{2}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{4}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.31) and (20.4.5).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.2

\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E2
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{3}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{3}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.32) and (20.4.4).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.3

\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E3
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>0$ because $1-\theta_{4}\left(0\middle|ix^{2}\right)=1-x^{-1}\theta_{2}\left(0\middle|ix^{% -2}\right)\sim 1$ as $x\to 0+$ , which follows from (20.7.33) and (20.4.3).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

… ►

§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

…

20: Bibliography O

… ►

F. Oberhettinger and T. P. Higgins (1961) Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

ⓘ

Referenced by:: §1.14(viii).
Encodings:: BibTeX

►

F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

ⓘ

External Links:: ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).
Encodings:: BibTeX

… ►

F. Oberhettinger (1972) Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.

ⓘ

Notes:: Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.
External Links:: ISBN 0-387-05997-0, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(v), §10.43(v), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §13.10(v), §13.23(iii), §15.14, §18.17(ix), §8.14, §8.6(iii).
Encodings:: BibTeX

… ►

F. Oberhettinger (1974) Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.

ⓘ

External Links:: ISBN 0-387-06942-9, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §20.10(i), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii).
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

…