explicit forms

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11: 3.8 Nonlinear Equations

… ►Sometimes the equation takes the form … ►

§3.8(iv) Zeros of Polynomials

… ►Explicit formulas for the zeros are available if

n\leq 4

; see §§1.11(iii) and 4.43. No explicit general formulas exist when

n\geq 5

. …

12: 18.35 Pollaczek Polynomials

… ►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8)) …or, equivalently in second form (18.2.10), …the recurrence relation of form (18.2.11_5) becomes … ►As in the coefficients of the above recurrence relations

n

and

c

only occur in the form

n+c

, the type 3 Pollaczek polynomials may also be called the associated type 2 Pollaczek polynomials by using the terminology of §18.30. … ►we have the explicit representations …

13: 18.38 Mathematical Applications

… ►A symmetric Laurent polynomial is a function of the form …where

Q_{0}

is a constant with explicit expression in terms of

e_{1},e_{2},e_{3},e_{4}

and

q

given in Koornwinder (2007a, (2.8)). … ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. … ►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures. …

14: 23.9 Laurent and Other Power Series

… ►Explicit coefficients

c_{n}

in terms of

c_{2}

and

c_{3}

are given up to

c_{19}

in Abramowitz and Stegun (1964, p. 636). … ►Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as

1/\wp\left(z\right)\to 0

. …

15: 8.11 Asymptotic Approximations and Expansions

… ►This reference also contains explicit formulas for

b_{k}(\lambda)

in terms of Stirling numbers and for the case

\lambda>1

an asymptotic expansion for

b_{k}(\lambda)

k\to\infty

. … ►

8.11.12

\Gamma\left(z,z\right)\sim z^{z-1}e^{-z}\left(\sqrt{\frac{\pi}{2}}z^{\frac{1}{% 2}}-\frac{1}{3}+\frac{\sqrt{2\pi}}{24z^{\frac{1}{2}}}-\frac{4}{135z}+\frac{% \sqrt{2\pi}}{576z^{\frac{3}{2}}}+\frac{8}{2835z^{2}}+\dots\right),

|\operatorname{ph}z|\leq 2\pi-\delta

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\operatorname{ph}$ : phase, $z$ : complex variable and $\delta$ : small positive constant
A&S Ref:: 6.5.35 (Modified form)
Referenced by:: §8.11(v), Firefox 3 users should upgrade, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E12
Encodings:: pMML, png, TeX
Addition (effective with 1.0.14):: The sector of validity for the above equation was increased from $|\operatorname{ph}z|\leq\pi-\delta$ to $|\operatorname{ph}z|\leq 2\pi-\delta$ .
See also:: Annotations for §8.11(v), §8.11 and Ch.8

… ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …

16: 4.13 Lambert $W$ -Function

… ►

4.13.3_1

W_{0}\left(x{\mathrm{e}}^{x}\right)=\begin{cases}x,&-1\leq x,\\ \text{(no simpler form)},&x<-1.\end{cases}

ⓘ

Symbols:: $W\left(\NVar{z}\right)$ : Lambert $W$ -function, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
Proof sketch:: Take $x=W_{0}\left(t\right)>-1$ , with $t>-{\mathrm{e}}^{-1}$ . Then $W_{0}\left(x{\mathrm{e}}^{x}\right)=W_{0}\left(W_{0}\left(t\right){\mathrm{e}}% ^{W_{0}\left(t\right)}\right)=W_{0}\left(t\right)=x$ .
Referenced by:: §4.13, Erratum (V1.1.7) for Expansion
Permalink:: http://dlmf.nist.gov/4.13.E3_1
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: This equation was added.
See also:: Annotations for §4.13 and Ch.4

►

4.13.3_2

W_{\pm 1}\left(x{\mathrm{e}}^{x}\mp 0\mathrm{i}\right)=\begin{cases}\text{(no % simpler form)},&-1\leq x,\\ x,&x<-1.\end{cases}

ⓘ

Symbols:: $W\left(\NVar{z}\right)$ : Lambert $W$ -function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $x$ : real variable
Proof sketch:: Take $x=W_{1}\left(t-0\mathrm{i}\right)<-1$ , with $-{\mathrm{e}}^{-1}<t<0$ . Then $W_{1}\left(x{\mathrm{e}}^{x}-0\mathrm{i}\right)=W_{1}\left(W_{1}\left(t-0% \mathrm{i}\right){\mathrm{e}}^{W_{1}\left(t-0\mathrm{i}\right)}-0\mathrm{i}% \right)=W_{1}\left(t-0\mathrm{i}\right)=x$ .
Referenced by:: §4.13, Erratum (V1.1.7) for Expansion
Permalink:: http://dlmf.nist.gov/4.13.E3_2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: This equation was added.
See also:: Annotations for §4.13 and Ch.4

… ►Explicit representations for the

p_{n}(x)

are given in Kalugin and Jeffrey (2011). …See Jeffrey and Murdoch (2017) for an explicit representation for the

c_{n}

in terms of associated Stirling numbers. …

17: 18.34 Bessel Polynomials

… ►Explicit (but complicated) weight functions

w(x)

taking both positive and negative values have been found such that (18.2.26) holds with

\,\mathrm{d}\mu(x)=w(x)\,\mathrm{d}x

; see Durán (1993), Evans et al. (1993), and Maroni (1995). … ►

18.34.8

\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)}_{n}\left(1+\alpha x\right)% }{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}\left(x;a\right).

ⓘ

Symbols:: $y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Bessel polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.34(iii)
Permalink:: http://dlmf.nist.gov/18.34.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.34(iii), §18.34 and Ch.18

…

18: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►An important asset of the distribution method is that it gives explicit expressions for the remainder terms associated with the resulting asymptotic expansions. … ►The distribution method outlined here can be extended readily to functions

f(t)

having an asymptotic expansion of the form … ►It is easily seen that

K^{+}

forms a commutative, associative linear algebra. … ►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …

19: Bibliography H

… ►

K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

… ►

F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

ⓘ

External Links:: ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(i).
Encodings:: BibTeX

… ►

J. H. Hubbard and B. B. Hubbard (2002) Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 2nd edition, Prentice Hall Inc., Upper Saddle River, NJ.

ⓘ

External Links:: ISBN 0-13-041408-5; 978-0-13-041408-3, MathReview (W. P. Ziemer), ZentralBlatt
Referenced by:: §1.6(i).
Encodings:: BibTeX

…

20: Bibliography G

… ►

E. T. Goodwin (1949b) The evaluation of integrals of the form

\int^{\infty}_{-\infty}f(x)e^{-x^{2}}dx

. Proc. Cambridge Philos. Soc. 45 (2), pp. 241–245.

ⓘ

External Links:: MathReview (S. Levy), ZentralBlatt
Referenced by:: §3.5(i).
Encodings:: BibTeX

… ►

H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

ⓘ

External Links:: FullText, MathReview (B. M. Stewart), ZentralBlatt
Referenced by:: §24.15(iii), §24.6.
Encodings:: BibTeX

… ►

C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.

ⓘ

External Links:: Document
Referenced by:: item Greene et al. (1979):, Notations F, Notations F, Notations G.
Encodings:: BibTeX

…