on intervals

SearchAdvancedHelp

(0.002 seconds)

31—40 of 237 matching pages

31: 24.1 Special Notation

… ► ►►►►

$j,k,\mathrm{\ell },m,n$	integers, nonnegative unless stated otherwise.
…
$(k,m)$	greatest common divisor of $k,m$.
$(k,m)=1$	$k$ and $m$ relatively prime.

…

32: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. … ►However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. …

33: 18.31 Bernstein–Szegő Polynomials

… ►The Bernstein–Szegő polynomials $\{p_n(x)\}$, $n=0,1,\mathrm{\dots }$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. …

34: 18.39 Applications in the Physical Sciences

… ►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty }).$ … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ … ►normalized with measure $r^{2}dr$, $r\in [0,\mathrm{\infty })$. … ►

Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applications^a

► ►►

Name of OP System	$w(x)$	$[a,b]$	Notation	Applications
…

►

… ►which maps $ϵ\in [0,\mathrm{\infty })$ onto $x\in [-1,1]$. …

35: 7.24 Approximations

… ►

Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\mathrm{erf}x$ on $0\le x\le 2$, for $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[2,\mathrm{\infty })$, and for ${\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[0,\mathrm{\infty })$ (30D).

►

Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x){\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $(0,\mathrm{\infty })$ (22D).

…

36: 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$(2,\mathbb{Z})$, …

37: 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). … ►A path ${𝐜}_1(t)$, $t\in [a,b]$, is a reparametrization of $𝐜(t^{\prime })$, $t^{\prime }\in [a^{\prime },b^{\prime }]$, if ${𝐜}_1(t)=𝐜(t^{\prime })$ and $t^{\prime }=h(t)$ with $h(t)$ differentiable and monotonic. … ►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. … ►For a surface of revolution, $y=f(x)$, $x\in [a,b]$, about the $x$-axis, …

38: 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. …

39: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $ℝ$. … ►

Hermite’s Differential Equation, $X=(-\mathrm{\infty },\mathrm{\infty })$

►The space $X$ is now the full real line, $(-\mathrm{\infty },\mathrm{\infty })$. … ►

Example 1: Bessel–Hankel Transform, $X=[0,\mathrm{\infty })$

… ►Pick $c\in (a,b)$. …

40: 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

…