# values at infinity

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##### 11: 16.2 Definition and Analytic Properties
The branch obtained by introducing a cut from $1$ to $+\infty$ on the real axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of ${{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$; 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 $\infty$. 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
(The fractional powers have their principal values.) … where the contour begins at $-\infty$, circles the origin once in the positive direction, and returns to $-\infty$. $t^{-z}$ has its principal value where $t$ crosses the positive real axis, and is continuous. … where $|\operatorname{ph}z|<\pi/2$ and the inverse tangent has its principal value. …
5.9.18 $\gamma=-\int_{0}^{\infty}e^{-t}\ln t\mathrm{d}t=\int_{0}^{\infty}\left(\frac{1% }{1+t}-e^{-t}\right)\frac{\mathrm{d}t}{t}=\int_{0}^{1}(1-e^{-t})\frac{\mathrm{% d}t}{t}-\int_{1}^{\infty}e^{-t}\frac{\mathrm{d}t}{t}=\int_{0}^{\infty}\left(% \frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\mathrm{d}t.$
##### 13: 15.6 Integral Representations
In (15.6.2) the point $\ifrac{1}{z}$ lies outside the integration contour, $t^{b-1}$ and $(t-1)^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\infty)$, and $(1-zt)^{a}=1$ at $t=0$. …
##### 14: 10.25 Definitions
###### §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 $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. … It has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. … 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
A sufficient condition for $p_{n}(x)$ to be the minimax polynomial is that $\left|\epsilon_{n}(x)\right|$ attains its maximum at $n+2$ distinct points in $[a,b]$ and $\epsilon_{n}(x)$ changes sign at these consecutive maxima. … There exists a unique solution of this minimax problem and there are at least $k+\ell+2$ values $x_{j}$, $a\leq x_{0}, such that $m_{j}=m$, where … 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 $r={\mathscr{L}}^{-1}R$. … Given $n+1$ distinct points $x_{k}$ in the real interval $[a,b]$, with ($a=$)$x_{0}($=b$), on each subinterval $[x_{k},x_{k+1}]$, $k=0,1,\ldots,n-1$, 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
This differential equation has a regular singularity at $z=0$ with indices $\pm\nu$, and an irregular singularity at $z=\infty$ of rank $1$; compare §§2.7(i) and 2.7(ii). … The principal branch of $J_{\nu}\left(z\right)$ corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\infty,0]$. … Each solution has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. 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 $(-\infty,0]$. …
##### 17: 6.16 Mathematical Applications
Hence, if $x$ is fixed and $n\to\infty$, then $S_{n}(x)\to\frac{1}{4}\pi$, $0$, or $-\frac{1}{4}\pi$ according as $0, $x=0$, or $-\pi; compare (6.2.14). … The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ for positive $x$ occurs at $x=\pi$ and equals $(1.1789\dots)\times\frac{1}{4}\pi$; compare Figure 6.3.2. Hence if $x=\pi/(2n)$ and $n\to\infty$, 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. … Figure 6.16.2: The logarithmic integral li ⁡ ( x ) , together with vertical bars indicating the value of π ⁡ ( x ) for x = 10 , 20 , … , 1000 . Magnify
##### 18: 25.12 Polylogarithms
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 $[1,\infty)$ and agrees with (25.12.1) when $|z|\leq 1$; see also §4.2(i). … Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ⁡ ( x + i ⁢ y ) | , - 20 ≤ x ≤ 20 , - 20 ≤ y ≤ 20 . Principal value. … Magnify 3D Help For other values of $z$, $\mathrm{Li}_{s}\left(z\right)$ is defined by analytic continuation. …
##### 19: 3.7 Ordinary Differential Equations
###### §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 $\mathscr{P}$ from $a$ to $b$, then $w(z)$ and $w^{\prime}(z)$ may be computed in a stable manner for $z=z_{0},z_{1},\dots,z_{P}$ by successive application of (3.7.5) for $j=0,1,\dots,P-1$, 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 $\mathscr{P}$, then we may reverse the labeling of the $z_{j}$ along $\mathscr{P}$ 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 $\mathscr{P}$ is at $\infty$, or if the differential equation is inhomogeneous. …
##### 20: 4.15 Graphics Figure 4.15.4: arctan ⁡ x and arccot ⁡ x . Only principal values are shown. … Magnify Figure 4.15.7 illustrates the conformal mapping of the strip $-\tfrac{1}{2}\pi<\Re z<\tfrac{1}{2}\pi$ onto the whole $w$-plane cut along the real axis from $-\infty$ to $-1$ and $1$ to $\infty$, where $w=\sin z$ and $z=\operatorname{arcsin}w$ (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 $(0,\infty)$. … In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …