# restricted position

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##### 1: 26.15 Permutations: Matrix Notation
where the sum is over $1\leq g and $n\geq h>\ell\geq 1$. … A permutation with restricted position specifies a subset $B\subseteq\{1,2,\ldots,n\}\times\{1,2,\ldots,n\}$. …
##### 2: 26.1 Special Notation
 $x$ real variable. … greatest common divisor of positive integers $h$ and $k$.
##### 4: 2.1 Definitions and Elementary Properties
(Here and elsewhere in this chapter $\delta$ is an arbitrary small positive constant.) … Let $\sum a_{s}x^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, … But for any given set of coefficients $a_{0},a_{1},a_{2},\dots$, and suitably restricted $\mathbf{X}$ there is an infinity of analytic functions $f(x)$ such that (2.1.14) and (2.1.16) apply. For (2.1.14) $\mathbf{X}$ can be the positive real axis or any unbounded sector in $\mathbb{C}$ of finite angle. …
##### 5: 11.6 Asymptotic Expansions
11.6.1 $\mathbf{K}_{\nu}\left(z\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{\Gamma% \left(k+\tfrac{1}{2}\right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\Gamma\left(\nu+\tfrac{% 1}{2}-k\right)},$ $|\operatorname{ph}z|\leq\pi-\delta$,
where $\delta$ is an arbitrary small positive constant. …If $\nu$ is real, $z$ is positive, and $m+\tfrac{1}{2}-\nu\geq 0$, then $R_{m}(z)$ is of the same sign and numerically less than the first neglected term. …
11.6.5 $\mathbf{H}_{\nu}\left(z\right),\mathbf{L}_{\nu}\left(z\right)\sim\frac{z}{\pi% \nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},$ $|\operatorname{ph}\nu|\leq\pi-\delta$.
11.6.9 $\mathbf{L}_{\nu}\left(\lambda\nu\right)\sim I_{\nu}\left(\lambda\nu\right),$ $|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta$,
##### 7: 13.21 Uniform Asymptotic Approximations for Large $\kappa$
When $\kappa\to\infty$ through positive real values with $\mu$ ($\geq 0$) fixed …uniformly with respect to $x\in(0,A]$ in each case, where $A$ is an arbitrary positive constant. Other types of approximations when $\kappa\to\infty$ through positive real values with $\mu$ ($\geq 0$) fixed are as follows. … uniformly with respect to $\mu\in[0,(1-\delta)\kappa]$ and $x\in\left(0,(1-\delta)(2\kappa+2\sqrt{\kappa^{2}-\mu^{2}})\right]$, where $\delta$ again denotes an arbitrary small positive constant. … This expansion is simpler in form than the expansions of Dunster (1989) that correspond to the approximations given in §13.21(iii), but the conditions on $\mu$ are more restrictive. …
##### 8: 1.10 Functions of a Complex Variable
and the integration contour is described once in the positive sense. … Here and elsewhere in this subsection the path $C$ is described in the positive sense. … where $N$ and $P$ are respectively the numbers of zeros and poles, counting multiplicity, of $f$ within $C$, and $\Delta_{C}(\operatorname{ph}f(z))$ is the change in any continuous branch of $\operatorname{ph}\left(f(z)\right)$ as $z$ passes once around $C$ in the positive sense. … (a) By introducing appropriate cuts from the branch points and restricting $F(z)$ to be single-valued in the cut plane (or domain). … The last condition means that given $\epsilon$ ($>0$) there exists a number $a_{0}\in[a,b)$ that is independent of $z$ and is such that …
##### 9: 15.12 Asymptotic Approximations
Let $\delta$ denote an arbitrary small positive constant. …
• (d)

$\Re z>\tfrac{1}{2}$ and $\alpha_{-}-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}c\leq\alpha_{+}+\tfrac{1% }{2}\pi-\delta$, where

15.12.1 $\alpha_{\pm}=\operatorname{arctan}\left(\frac{\operatorname{ph}z-\operatorname% {ph}\left(1-z\right)\mp\pi}{\ln|1-z^{-1}|}\right),$

with $z$ restricted so that $\pm\alpha_{\pm}\in[0,\tfrac{1}{2}\pi)$.

• Again, throughout this subsection $\delta$ denotes an arbitrary small positive constant, and $a,b,c,z$ are real or complex and fixed. …
##### 10: Mathematical Introduction
This is because $\mathbf{F}$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $\mathbf{F}$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. …
 $(a,b]$ or $[a,b)$ half-closed intervals. … $m\equiv n\pmod{p}$ means $p$ divides $m-n$, where $m$, $n$, and $p$ are positive integers with $m>n$. set of all positive integers. …