case λ=0

(0.004 seconds)

31—40 of 128 matching pages

31: 4.13 Lambert $W$ -Function

… ►The other branches

W_{k}\left(z\right)

are single-valued analytic functions on

\mathbb{C}\setminus(-\infty,0]

, have a logarithmic branch point at

z=0

, and, in the case

k=\pm 1

, have a square root branch point at

z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}

respectively. … ►In the case of

k=0

and real

z

the series converges for

z\geq\mathrm{e}

. …

32: 6.7 Integral Representations

… ►The first integrals on the right-hand sides apply when

|\operatorname{ph}z|<\pi

; the second ones when

\Re z\geq 0

and (in the case of (6.7.14))

z\neq 0

. …

33: 5.11 Asymptotic Expansions

… ►For the error term in (5.11.19) in the case

z=x\;(>0)

and

c=1

, see Olver (1995). …

34: 18.17 Integrals

… ►Formula (18.17.9), after substitution of (18.5.7), is a special case of (15.6.8). Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of

n

-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … ►In particular, in case of exponential Fourier transforms, we may assume

y\in\mathbb{R}

. … ►Formulas (18.17.21_2) and (18.17.21_3) are respectively the limit case

c\to\tfrac{1}{2}

and the special case

c=1

of (18.17.21_1). … ►The case

x=1

is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972). …

35: 23.22 Methods of Computation

… ►

(a)

In the general case, given by $cd\neq 0$ , we compute the roots $\alpha$ , $\beta$ , $\gamma$ , say, of the cubic equation $4t^{3}-ct-d=0$ ; see §1.11(iii). These roots are necessarily distinct and represent $e_{1}$ , $e_{2}$ , $e_{3}$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^{3}-27d^{2}>0$ , then $e_{1}$ , $e_{2}$ , $e_{3}$ can be identified via (23.5.1), and $k^{2}$ , ${k^{\prime}}^{2}$ obtained from (23.6.16).

If $c^{3}-27d^{2}<0$ , or $c$ and $d$ are not both real, then we label $\alpha$ , $\beta$ , $\gamma$ so that the triangle with vertices $\alpha$ , $\beta$ , $\gamma$ is positively oriented and $[\alpha,\gamma]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$ , $\beta$ , $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha,\gamma)$ . In consequence, $k^{2}=(\beta-\gamma)/(\alpha-\gamma)$ , ${k^{\prime}}^{2}=(\alpha-\beta)/(\alpha-\gamma)$ satisfy $\Im k^{2}\geq 0\geq\Im{k^{\prime}}^{2}$ (with strict inequality unless $\alpha$ , $\beta$ , $\gamma$ are collinear); also $|k^{2}|$ , $|{k^{\prime}}^{2}|\leq 1$ .

Finally, on taking the principal square roots of $k^{2}$ and ${k^{\prime}}^{2}$ we obtain values for $k$ and $k^{\prime}$ that lie in the 1st and 4th quadrants, respectively, and $2\omega_{1}$ , $2\omega_{3}$ are given by

23.22.1

2\omega_{1}M\left(1,k^{\prime}\right)=-2i\omega_{3}M\left(1,k\right)=\frac{\pi% }{3}\sqrt{\frac{c(2+k^{2}{k^{\prime}}^{2})({k^{\prime}}^{2}-k^{2})}{d(1-k^{2}{% k^{\prime}}^{2})}},

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
Proof sketch:: Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.
Referenced by:: §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ( $2\omega_{1}$ , $2\omega_{3}$ ), corresponding to the 2 possible choices of the square root.

…

36: 2.9 Difference Equations

… ►(For the case

g_{0}=0

see the final paragraph of §2.9(ii) with

Q

negative.) …

37: Errata

… ►

Equation (18.7.25)

18.7.25

\lim_{\lambda\to 0}\frac{n+\lambda}{\lambda}C^{(\lambda)}_{n}\left(x\right)=% \begin{cases}1,&\text{$n=0$,}\\ 2T_{n}\left(x\right),&\text{$n=1,2,\dots$}\end{cases}

We included the case $n=0$ .

… ►

Equation (18.35.2)

18.35.2

P^{{(\lambda)}}_{n+1}\left(x;a,b,c\right)=\frac{2(n+c+\lambda+a)x+2b}{n+c+1}\,% P^{{(\lambda)}}_{n}\left(x;a,b,c\right)-\frac{n+c+2\lambda-1}{n+c+1}\,P^{{(% \lambda)}}_{n-1}\left(x;a,b,c\right),

n=0,1,\dots

This recurrence relation which was previously given for Pollaczek polynomials of type 2 (the case $c=0$ ) has been updated for Pollaczek polynomials of type 3.

►

Equation (18.35.5)

18.35.5

\int_{-1}^{1}P^{(\lambda)}_{n}\left(x;a,b\right)P^{(\lambda)}_{m}\left(x;a,b% \right)w^{(\lambda)}(x;a,b)\,\mathrm{d}x=\frac{\Gamma\left(2\lambda+n\right)}{% n!\,(\lambda+a+n)}\,\delta_{n,m},

a\geq b\geq-a

\lambda>0

This equation was updated to give the full normalization. Previously the constraints on $a$ , $b$ and $\lambda$ were given in (18.35.6) and included $\lambda>-\frac{1}{2}$ . The case $-\frac{1}{2}<\lambda\leq 0$ is now discussed in (18.35.6_2)–(18.35.6_4).

… ►

Paragraph Mellin–Barnes Integrals (in §8.6(ii))

The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$ . The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma\left(a,z\right)$ . In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\ldots$ .

Reported 2017-07-10 by Kurt Fischer.

…

38: 18.35 Pollaczek Polynomials

… ►Hence, only in the case

a=b=0

does

\ln\left(w^{(\lambda)}(x;a,b)\right)

satisfy the condition (18.2.39) for the Szegő class

\mathcal{G}

. … ►

18.35.6_2

\mathrm{(i)}\;\lambda>0\mbox{ and }a+\lambda>0,\quad\mathrm{(ii)}\;-\tfrac{1}{% 2}<\lambda<0\mbox{ and }-1<a+\lambda<0,\quad\mathrm{(iii)}\;\lambda=0\mbox{ % and }a=b=0.

ⓘ

Symbols:: $n$ : nonnegative integer and $x$ : real variable
Proved:: Ismail (2009, (5.5.8)); Case (iii) is overlooked in the given reference.(proved)
Proof sketch:: By Favard’s theorem (see §18.2(viii)) the necessary and sufficient condition for orthogonality is that in (18.35.2_4), for $c=0$ , the coefficient of $Q^{(\lambda)}_{n}(x;a,b,0)$ is real for $n\geq 0$ and the coefficient of $Q^{(\lambda)}_{n-1}(x;a,b,0)$ is positive for $n\geq 1$ .
Referenced by:: §18.35(ii), Erratum (V1.2.0) for Equation (18.35.5)
Permalink:: http://dlmf.nist.gov/18.35.E6_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.35(ii), §18.35 and Ch.18

…

39: 18.5 Explicit Representations

… ►Similarly in the cases of the ultraspherical polynomials

C^{(\lambda)}_{n}\left(x\right)

and the Laguerre polynomials

L^{(\alpha)}_{n}\left(x\right)

we assume that

\lambda>-\tfrac{1}{2},\lambda\neq 0

, and

\alpha>-1

, unless stated otherwise. …

40: 18.9 Recurrence Relations and Derivatives

… ►with initial values

p_{0}(x)=1

and

p_{1}(x)=A_{0}x+B_{0}

. … ►

A_{0}

and

B_{0}

have to be understood for

\alpha+\beta=0

-1

by continuity in

\alpha

and

\beta

, that is,

A_{0}=\tfrac{1}{2}(\alpha+\beta)+1

and

B_{0}=\tfrac{1}{2}(\alpha-\beta)

. … ►with initial values

p_{0}(x)=1

and

p_{1}(x)=a_{0}^{-1}(x-b_{0})

. … ►Formulas (18.9.5), (18.9.11), (18.9.13) are special cases of (18.2.16). Formulas (18.9.6), (18.9.12), (18.9.14) are special cases of (18.2.17). …