Richmond () take Driver License【仿证微CXFK69】H12eGSA

SearchAdvancedHelp

The terms "H12eGSA", "cxfk69" were not found.

(0.004 seconds)

1—10 of 88 matching pages

1: Joyce E. Conlon

… ► 1957 in Richmond, Virginia, d. …

2: Bibliography C

… ►

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

… ►

Algorithm 715: SPECFUN – A portable FORTRAN package of special function routines and test drivers. ACM Trans. Math. Software 19 (1), pp. 22–32.

ⓘ

Notes:

Double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, Remark. GAMS (module 715; package TOMS), ZentralBlatt

Referenced by:

3rd item, 6th item, 4th item, 2nd item, 2nd item, 2nd item, 2nd item.

Encodings:

BibTeX

…

3: 18.27 $q$-Hahn Class

… ►

18.27.12_5 $$\underset{q\to 1}{lim}{P}_n^{(\alpha ,\beta )}(x;c,d;q)={\left(\frac{c+d}{2}\right)}^n{P}_n^{(\alpha ,\beta )}\left(\frac{2x-c+d}{c+d}\right).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Andrews and Askey (1985, (3.33)); for $c=d=1$

Proof sketch:

On the left substitute first (18.27.6) and next (18.27.5). Then, after taking the limit, a ${}_2{}^{}F_1^{}$ hypergeometric function occurs. Finally use (18.5.7).

Referenced by:

§18.27(iii), §18.27(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E12_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iii), §18.27(iii), §18.27 and Ch.18

… ►Bounds for the extreme zeros are given in Driver and Jordaan (2013). … ►Bounds for the extreme zeros are given in Driver and Jordaan (2013). …

4: 12.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …

5: Bille C. Carlson

… ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. …

6: About Color Map

… ►We therefore use a piecewise linear mapping as illustrated below, that takes phase $0$ to red, $\pi /2$ to yellow, $\pi$ to cyan and $3\pi /2$ to blue. …

7: 2.2 Transcendental Equations

… ►We may take $a=\frac{1}{2}$. …

8: 8.15 Sums

… ►

8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

…

9: 8.27 Approximations

… ►

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

…

10: 18.16 Zeros

… ►See Dimitrov and Nikolov (2010), and Driver and Jordaan (2013). … ►See Driver and Jordaan (2013). …