# norms

(0.001 seconds)

## 1—10 of 406 matching pages

##### 1: 35.11 Tables
Tables of zonal polynomials are given in James (1964) for $|\kappa|\leq 6$, Parkhurst and James (1974) for $|\kappa|\leq 12$, and Muirhead (1982, p. 238) for $|\kappa|\leq 5$. …
##### 2: 10.37 Inequalities; Monotonicity
If $0\leq\nu<\mu$ and $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$, then Note that previously we did mention that (10.37.1) holds for $|\operatorname{ph}z|<\pi$. …
##### 3: 20.1 Special Notation
 $m$, $n$ integers. … the nome, $q=e^{i\pi\tau}$, $0<\left|q\right|<1$. Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\Re\tau=0$, $\Im\tau>0$, so that $0 and $\tau$ and $q$ are uniquely related. …
The main functions treated in this chapter are the theta functions $\theta_{j}\left(z\middle|\tau\right)=\theta_{j}\left(z,q\right)$ where $j=1,2,3,4$ and $q=e^{i\pi\tau}$. … Jacobi’s original notation: $\Theta(z|\tau)$, $\Theta_{1}(z|\tau)$, $\mathrm{H}(z|\tau)$, $\mathrm{H}_{1}(z|\tau)$, respectively, for $\theta_{4}\left(u\middle|\tau\right)$, $\theta_{3}\left(u\middle|\tau\right)$, $\theta_{1}\left(u\middle|\tau\right)$, $\theta_{2}\left(u\middle|\tau\right)$, where $u=z/{\theta_{3}}^{2}\left(0\middle|\tau\right)$. … Neville’s notation: $\theta_{s}(z|\tau)$, $\theta_{c}(z|\tau)$, $\theta_{d}(z|\tau)$, $\theta_{n}(z|\tau)$, respectively, for ${\theta_{3}}^{2}\left(0\middle|\tau\right)\ifrac{\theta_{1}\left(u\middle|\tau% \right)}{\theta_{1}'\left(0\middle|\tau\right)}$, $\ifrac{\theta_{2}\left(u\middle|\tau\right)}{\theta_{2}\left(0\middle|\tau% \right)}$, $\ifrac{\theta_{3}\left(u\middle|\tau\right)}{\theta_{3}\left(0\middle|\tau% \right)}$, $\ifrac{\theta_{4}\left(u\middle|\tau\right)}{\theta_{4}\left(0\middle|\tau% \right)}$, where again $u=z/{\theta_{3}}^{2}\left(0\middle|\tau\right)$. … McKean and Moll’s notation: $\vartheta_{j}(z|\tau)=\theta_{j}\left(\pi z\middle|\tau\right)$, $j=1,2,3,4$. …
##### 4: Sidebar 7.SB1: Diffraction from a Straightedge
The intensity distribution follows $|\mathcal{F}\left(x\right)|^{2}$, where $\mathcal{F}$ is the Fresnel integral (See 7.3.4). …
##### 5: 4.18 Inequalities
4.18.9 $|\sin z|\leq\sinh|z|,$
$|\cos z|<2,$
$|\sin z|\leq\tfrac{6}{5}|z|$ , $|z|<1$.
For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. If $p$ divides $n$, then the value of $(n|p)$ is $0$. …The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). … If an odd integer $P$ has prime factorization $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …
##### 7: 20.2 Definitions and Periodic Properties
Corresponding expansions for $\theta_{j}'\left(z\middle|\tau\right)$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. … For fixed $\tau$, each $\theta_{j}\left(z\middle|\tau\right)$ is an entire function of $z$ with period $2\pi$; $\theta_{1}\left(z\middle|\tau\right)$ is odd in $z$ and the others are even. For fixed $z$, each of $\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}$, $\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}$, $\theta_{3}\left(z\middle|\tau\right)$, and $\theta_{4}\left(z\middle|\tau\right)$ is an analytic function of $\tau$ for $\Im\tau>0$, with a natural boundary $\Im\tau=0$, and correspondingly, an analytic function of $q$ for $\left|q\right|<1$ with a natural boundary $\left|q\right|=1$. … For $m,n\in\mathbb{Z}$, the $z$-zeros of $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, are $(m+n\tau)\pi$, $(m+\tfrac{1}{2}+n\tau)\pi$, $(m+\tfrac{1}{2}+(n+\tfrac{1}{2})\tau)\pi$, $(m+(n+\tfrac{1}{2})\tau)\pi$ respectively.
##### 8: 27.6 Divisor Sums
27.6.1 $\sum_{d\mathbin{|}n}\lambda\left(d\right)=\begin{cases}1,&n\mbox{ is a square}% ,\\ 0,&\mbox{otherwise}.\end{cases}$
27.6.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)f(d)=\prod_{p\mathbin{|}n}(1-f(p)),$ $n>1$.
27.6.4 $\sum_{d^{2}\mathbin{|}n}\mu\left(d\right)=|\mu\left(n\right)|,$
##### 9: 27.7 Lambert Series as Generating Functions
If $|x|<1$, then the quotient $x^{n}/(1-x^{n})$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series:
27.7.2 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sum_{d% \mathbin{|}n}f(d)x^{n}.$
Again with $|x|<1$, special cases of (27.7.2) include:
27.7.3 $\sum_{n=1}^{\infty}\mu\left(n\right)\frac{x^{n}}{1-x^{n}}=x,$
27.7.4 $\sum_{n=1}^{\infty}\phi\left(n\right)\frac{x^{n}}{1-x^{n}}=\frac{x}{(1-x)^{2}},$