# values at infinity

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## 11—20 of 119 matching pages

##### 11: 16.2 Definition and Analytic Properties

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►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty}$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi $, is the

*principal branch*(or*principal value*) of ${}_{q+1}F_{q}(\mathbf{a};\mathbf{b};z)$; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty}$.*Unless indicated otherwise*it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … ►In general the series (16.2.1) diverges for all nonzero values of $z$. … ►(However,*except where indicated otherwise*in the DLMF we assume that when $p>q+1$ at least one of the ${a}_{k}$ is a nonpositive integer.) …##### 12: 5.9 Integral Representations

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►(The fractional powers have their principal values.)
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►where the contour begins at
$-\mathrm{\infty}$, circles the origin once in the positive direction, and returns to $-\mathrm{\infty}$.
${t}^{-z}$ has its principal value where $t$ crosses the positive real axis, and is continuous.
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►where $$ and the inverse tangent has its principal value.
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5.9.18
$$\gamma =-{\int}_{0}^{\mathrm{\infty}}{\mathrm{e}}^{-t}\mathrm{ln}tdt={\int}_{0}^{\mathrm{\infty}}\left(\frac{1}{1+t}-{\mathrm{e}}^{-t}\right)\frac{dt}{t}={\int}_{0}^{1}(1-{\mathrm{e}}^{-t})\frac{dt}{t}-{\int}_{1}^{\mathrm{\infty}}{\mathrm{e}}^{-t}\frac{dt}{t}={\int}_{0}^{\mathrm{\infty}}\left(\frac{{\mathrm{e}}^{-t}}{1-{\mathrm{e}}^{-t}}-\frac{{\mathrm{e}}^{-t}}{t}\right)dt.$$

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##### 13: 15.6 Integral Representations

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►In (15.6.2) the point $1/z$ lies outside the integration contour, ${t}^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty})$, and ${(1-zt)}^{a}=1$
at
$t=0$.
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##### 14: 10.25 Definitions

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###### §10.25(ii) Standard Solutions

… ►In particular, the*principal branch*of ${I}_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu}$, is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. … ►It has a branch point at $z=0$ for all $\nu \in \u2102$. The*principal branch*corresponds to the principal value of the square root in (10.25.3), is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. … ►*Except where indicated otherwise*it is assumed throughout the DLMF that the symbols ${I}_{\nu}\left(z\right)$ and ${K}_{\nu}\left(z\right)$ denote the principal values of these functions. …##### 15: 3.11 Approximation Techniques

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►A sufficient condition for ${p}_{n}(x)$ to be the minimax polynomial is that $\left|{\u03f5}_{n}(x)\right|$ attains its maximum at
$n+2$ distinct points in $[a,b]$ and ${\u03f5}_{n}(x)$ changes sign at these consecutive maxima.
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►There exists a unique solution of this minimax problem and there are at least $k+\mathrm{\ell}+2$
values
${x}_{j}$, $$, such that ${m}_{j}=m$, where
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►is called a

*Padé approximant at zero*of $f$ if … ►A general procedure is to approximate $F$ by a rational function $R$ (vanishing at infinity) and then approximate $f$ by $r={\mathcal{L}}^{-1}R$. … ►Given $n+1$ distinct points ${x}_{k}$ in the real interval $[a,b]$, with ($a=$)$$($=b$), on each subinterval $[{x}_{k},{x}_{k+1}]$, $k=0,1,\mathrm{\dots},n-1$, a low-degree polynomial is defined with coefficients determined by, for example, values ${f}_{k}$ and ${f}_{k}^{\prime}$ of a function $f$ and its derivative at the nodes ${x}_{k}$ and ${x}_{k+1}$. …##### 16: 10.2 Definitions

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►This differential equation has a regular singularity at
$z=0$ with indices $\pm \nu $, and an irregular singularity at
$z=\mathrm{\infty}$ of rank $1$; compare §§2.7(i) and 2.7(ii).
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►The

*principal branch*of ${J}_{\nu}\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu}$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty},0]$. … ►Each solution has a branch point at $z=0$ for all $\nu \in \u2102$. The*principal branches*correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty},0]$. … ► …##### 17: 6.16 Mathematical Applications

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►Hence, if $x$ is fixed and $n\to \mathrm{\infty}$, then ${S}_{n}(x)\to \frac{1}{4}\pi $, $0$, or $-\frac{1}{4}\pi $ according as $$, $x=0$, or $$; compare (6.2.14).
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►The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ for positive $x$ occurs at
$x=\pi $ and equals $(1.1789\mathrm{\dots})\times \frac{1}{4}\pi $; compare Figure 6.3.2.
Hence if $x=\pi /(2n)$ and $n\to \mathrm{\infty}$, then the limiting value of ${S}_{n}(x)$ overshoots $\frac{1}{4}\pi $ by approximately 18%.
Similarly if $x=\pi /n$, then the limiting value of ${S}_{n}(x)$ undershoots $\frac{1}{4}\pi $ by approximately 10%, and so on.
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##### 18: 25.12 Polylogarithms

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►Other notations and names for ${\mathrm{Li}}_{2}\left(z\right)$ include ${S}_{2}(z)$ (Kölbig et al. (1970)), Spence function $\mathrm{Sp}(z)$ (’t Hooft and Veltman (1979)), and ${\mathrm{L}}_{2}(z)$ (Maximon (2003)).
►In the complex plane ${\mathrm{Li}}_{2}\left(z\right)$ has a branch point at
$z=1$.
The principal branch has a cut along the interval $[1,\mathrm{\infty})$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i).
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►For other values of $z$, ${\mathrm{Li}}_{s}\left(z\right)$ is defined by analytic continuation.
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##### 19: 3.7 Ordinary Differential Equations

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###### §3.7(ii) Taylor-Series Method: Initial-Value Problems

… ►If the solution $w(z)$ that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along $\mathcal{P}$ from $a$ to $b$, then $w(z)$ and ${w}^{\prime}(z)$ may be computed in a stable manner for $z={z}_{0},{z}_{1},\mathrm{\dots},{z}_{P}$ by successive application of (3.7.5) for $j=0,1,\mathrm{\dots},P-1$, beginning with initial values $w(a)$ and ${w}^{\prime}(a)$. … ►Similarly, if $w(z)$ is decaying at least as fast as all other solutions along $\mathcal{P}$, then we may reverse the labeling of the ${z}_{j}$ along $\mathcal{P}$ and begin with initial values $w(b)$ and ${w}^{\prime}(b)$. ►###### §3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►The latter is especially useful if the endpoint $b$ of $\mathcal{P}$ is at $\mathrm{\infty}$, or if the differential equation is inhomogeneous. …##### 20: 4.15 Graphics

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►Figure 4.15.7 illustrates the conformal mapping of the strip $$ onto the whole $w$-plane cut along the real axis from $-\mathrm{\infty}$ to $-1$ and $1$ to $\mathrm{\infty}$, where $w=\mathrm{sin}z$ and $z=\mathrm{arcsin}w$ (principal value).
…Lines parallel to the real axis in the $z$-plane map onto ellipses in the $w$-plane with foci at
$w=\pm 1$, and lines parallel to the imaginary axis in the $z$-plane map onto rectangular hyperbolas confocal with the ellipses.
In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty})$.
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►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase.
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