open

SearchAdvancedHelp

(0.001 seconds)

21—30 of 195 matching pages

21: 14.21 Definitions and Basic Properties

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). The principal branches of $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are real when $\nu$, $\mu \in ℝ$ and $z\in (1,\mathrm{\infty })$. … ►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. …

22: 18.16 Zeros

… ►Let ${\theta }_{n,m}={\theta }_{n,m}^{(\alpha ,\beta )}$, $m=1,2,\mathrm{\dots },n$, denote the zeros of $P_n^{(\alpha ,\beta )}\left(\cos \theta \right)$ as function of $\theta$ with … ►

18.16.2 $${\theta }_{n,m}^{(-\frac{1}{2},\frac{1}{2})}=\frac{(m-\frac{1}{2})\pi }{n+\frac{1}{2}}\le {\theta }_{n,m}^{(\alpha ,\beta )}\le \frac{m\pi }{n+\frac{1}{2}}={\theta }_{n,m}^{(\frac{1}{2},-\frac{1}{2})},$$ $\alpha ,\beta \in [-\frac{1}{2},\frac{1}{2}]$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and ${\theta }_{n,m}$: zeros

Referenced by:

§18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3)

Permalink:

http://dlmf.nist.gov/18.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

►

18.16.3 $${\theta }_{n,m}^{(-\frac{1}{2},-\frac{1}{2})}=\frac{(m-\frac{1}{2})\pi }{n}\le {\theta }_{n,m}^{(\alpha ,\alpha )}\le \frac{m\pi }{n+1}={\theta }_{n,m}^{(\frac{1}{2},\frac{1}{2})},$$ $\alpha \in [-\frac{1}{2},\frac{1}{2}]$, $m=1,2,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $(a,b)$: open interval, $m$: nonnegative integer, $n$: nonnegative integer and ${\theta }_{n,m}$: zeros

Referenced by:

§18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3)

Permalink:

http://dlmf.nist.gov/18.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

… ►when $\alpha \notin (-\frac{1}{2},\frac{1}{2})$. … ►All zeros of $H_n\left(x\right)$ lie in the open interval $(-\sqrt{2n+1},\sqrt{2n+1})$. …

23: 23.15 Definitions

… ►The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, …

24: 8.13 Zeros

… ►

Table 8.13.1: Double zeros $(a_n^{\ast },x_n^{\ast })$ of ${\gamma }^{\ast }(a,x)$.

► ►►

$n$	$a_n^{\ast }$		$x_n^{\ast }$
…

►

25: 1.6 Vectors and Vector-Valued Functions

… ►Note: The terminology open and closed sets and boundary points in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►with $(u,v)\in D$, an open set in the plane. … ►The vector ${𝐓}_u\times {𝐓}_v$ at $(u_0,v_0)$ is normal to the surface at $𝚽(u_0,v_0)$. …

26: 22.18 Mathematical Applications

… ►The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. … ►For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …

27: 26.9 Integer Partitions: Restricted Number and Part Size

… ►It is also equal to the number of lattice paths from $(0,0)$ to $(m,k)$ that have exactly $n$ vertices $(h,j)$, $1\le h\le m$, $1\le j\le k$, above and to the left of the lattice path. …

28: 5.3 Graphics

… ►

►►

Figure 5.3.2: $\ln \mathrm{\Gamma }\left(x\right)$. This function is convex on $(0,\mathrm{\infty })$; compare §5.5(iv). Magnify

…

29: 18.2 General Orthogonal Polynomials

… ►Let $(a,b)$ be a finite or infinite open interval in $ℝ$. … ►Assume $y\notin (a,b)$ in (18.2.12). … ►(convergence in $L_w^2((a,b))$). … ►This is the class of weight functions $w$ on $(-1,1)$ such that, in addition to (18.2.1_5), … ►

The Nevai class $𝐌(a,b)$

…

30: 17.12 Bailey Pairs

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ … ►If $({\alpha }_n,{\beta }_n)$ is a Bailey pair, then … ►If $({\alpha }_n,{\beta }_n)$ is a Bailey pair, then so is $({\alpha }_n^{\prime },{\beta }_n^{\prime })$, where …