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11: 18.40 Methods of Computation

… ►Let $x^{\prime }\in (a,b)$. … ►Here $x(t,N)$ is an interpolation of the abscissas $x_{i,N},i=1,2,\mathrm{\dots },N$, that is, $x(i,N)=x_{i,N}$, allowing differentiation by $i$. … ►

18.40.9 $$x(t,N)=\frac{x_{1,N}}{1+}\frac{a_1(t-1)}{1+}\frac{a_2(t-2)}{1+}\mathrm{\cdots }\frac{a_{N-1}(t-(N-1))}{1},$$ $t\in (0,\mathrm{\infty })$,

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Symbols:

$\in$: element of, $(a,b)$: open interval, $N$: positive integer, $t$: real variable and $x$: real variable

Source:

Schlessinger (1968)

Permalink:

http://dlmf.nist.gov/18.40.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►The PWCF $x(t,N)$ is a minimally oscillatory algebraic interpolation of the abscissas $x_{i,N},i=1,2,\mathrm{\dots },N$. … ►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

12: 28.17 Stability as $x\to \pm \mathrm{\infty }$

§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. For example, as $x\to +\mathrm{\infty }$ one of the solutions ${\mathrm{me}}_{\nu }(x,q)$ and ${\mathrm{me}}_{\nu }(-x,q)$ tends to $0$ and the other is unbounded (compare Figure 28.13.5). … ►For real $a$ and $q$ $(\ne 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. …

13: 1.4 Calculus of One Variable

… ►If $f(x)$ is continuous at each point $c\in (a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in C(a,b)$. … ►If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^n(I)$. … ►Then for $f(x)$ continuous on $(a,b)$, … ►If , then $f(x)$ is of bounded variation on $(a,b)$. … ►A function $f(x)$ is convex on $(a,b)$ if …

14: 18.1 Notation

… ► ►►►

$x,y,t$	real variables.
…
$w(x)$	weight function $(\ge 0)$ on an open interval $(a,b)$.
…

… ►

$${(z;q)}_{\mathrm{\infty }}=\prod _{j=0}^{\mathrm{\infty }}(1-z{q}^j),$$

►

$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

… ►

Disk: $R_{m,n}^{(\alpha )}\left(z\right)$.

…

15: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . … ►with $(s_1,s_2)\in \{0,1,a,\mathrm{\infty }\}$, denotes a set of solutions of (31.2.1), each of which is analytic at $s_1$ and $s_2$. …

16: 4.23 Inverse Trigonometric Functions

… ►The function ${(1-t^2)}^{1/2}$ assumes its principal value when $t\in (-1,1)$; elsewhere on the integration paths the branch is determined by continuity. … ►

4.23.19 $$\arcsin z=-\mathrm{i}\ln \left({(1-z^2)}^{1/2}+\mathrm{i}z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1)\cup (1,\mathrm{\infty })$;

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arcsin z$: arcsine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Referenced by:

§4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E19

Encodings:

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See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

… ►

4.23.26 $$\arctan z=\frac{\mathrm{i}}{2}\ln \left(\frac{\mathrm{i}+z}{\mathrm{i}-z}\right),$$ $z/\mathrm{i}\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Referenced by:

§4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

… ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. … ►where $z=x+\mathrm{i}y$ and $\pm z\notin (1,\mathrm{\infty })$ in (4.23.34) and (4.23.35), and in (4.23.36). …

17: 18.3 Definitions

… ►

With the property that ${\{p_{n+1}^{\prime }(x)\}}_{n=0}^{\mathrm{\infty }}$ is again a system of OP’s. See §18.9(iii).

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

► ►►►►

Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…
Hermite	$H_n\left(x\right)$	$(-\mathrm{\infty },\mathrm{\infty })$	${\mathrm{e}}^{-x^2}$	${\pi }^{\frac{1}{2}}{2}^{n}n!$	$2^n$	$0$
Hermite	${\mathrm{𝐻𝑒}}_n\left(x\right)$	$(-\mathrm{\infty },\mathrm{\infty })$	${\mathrm{e}}^{-\frac{1}{2}{x}^2}$	${(2\pi )}^{\frac{1}{2}}n!$	$1$	$0$

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… ►For $-1-\beta >\alpha >-1$ a finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ is orthogonal on $(1,\mathrm{\infty })$ with weight function $w(x)={(x-1)}^{\alpha }{(x+1)}^{\beta }$. For $\nu \in ℝ$ and $N>-\frac{1}{2}$ a finite system of Jacobi polynomials $P_n^{(-N-1+\mathrm{i}\nu ,-N-1-\mathrm{i}\nu )}\left(\mathrm{i}x\right)$ (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on $(-\mathrm{\infty },\mathrm{\infty })$ with $w(x)={\left(1+x^2\right)}^{-N-1}{\mathrm{e}}^{2\nu \arctan x}$. …

18: 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).

…

19: 23.20 Mathematical Applications

… ►Points $P=(x,y)$ on the curve can be parametrized by $x=\mathrm{\wp }(z;g_2,g_3)$, $2y={\mathrm{\wp }}^{\prime }(z;g_2,g_3)$, where $g_2=-4a$ and $g_3=-4b$: in this case we write $P=P(z)$. The curve $C$ is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element $o=(0,1,0)$ as the point at infinity, the negative of $P=(x,y)$ by $-P=(x,-y)$, and generally $P_1+P_2+P_3=0$ on the curve iff the points $P_1$, $P_2$, $P_3$ are collinear. … ►In terms of $(x,y)$ the addition law can be expressed $(x,y)+o=(x,y)$, $(x,y)+(x,-y)=o$; otherwise $(x_1,y_1)+(x_2,y_2)=(x_3,y_3)$, where … ► $K$ always has the form $T\times {\mathbb{Z}}^r$ (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the rank of $K$, raises questions of great difficulty, many of which are still open. …To determine $T$, we make use of the fact that if $(x,y)\in T$ then $y^2$ must be a divisor of $\mathrm{\Delta }$; hence there are only a finite number of possibilities for $y$. …

20: 14.21 Definitions and Basic Properties

… ► $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. 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 $ℂ\backslash (-\mathrm{\infty },1]$. …