# induced modulus

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## 3 matching pages

##### 1: 27.8 Dirichlet Characters
A divisor $d$ of $k$ is called an induced modulus for $\chi$ if … where $\chi_{0}$ is a character (mod $d$) for some induced modulus $d$ for $\chi$, and $\chi_{1}$ is the principal character (mod $k$). …
##### 2: 19.33 Triaxial Ellipsoids
The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength $H/(1+L_{c}\chi)$, where $L_{c}$ is the demagnetizing factor, given in cgs units by …
##### 3: 19.25 Relations to Other Functions
then the five nontrivial permutations of $x,y,z$ that leave $R_{F}$ invariant change $k^{2}$ ($=(z-y)/(z-x)$) into $1/k^{2}$, ${k^{\prime}}^{2}$, $1/{k^{\prime}}^{2}$, $-k^{2}/{k^{\prime}}^{2}$, $-{k^{\prime}}^{2}/k^{2}$, and $\sin\phi$ ($=\sqrt{(z-x)/z}$) into $k\sin\phi$, $-i\tan\phi$, $-ik^{\prime}\tan\phi$, $(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin^{2}}\phi}$, $-ik\sin\phi/\sqrt{1-k^{2}{\sin^{2}}\phi}$. Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … Let $r=1/x^{2}$. …
$\phi=\operatorname{arccos}\sqrt{\ifrac{x}{z}}=\operatorname{arcsin}\sqrt{% \ifrac{(z-x)}{z}},$
where we assume $0\leq x^{2}\leq 1$ if $x=\operatorname{sn}$, $\operatorname{cn}$, or $\operatorname{cd}$; $x^{2}\geq 1$ if $x=\operatorname{ns}$, $\operatorname{nc}$, or $\operatorname{dc}$; $x$ real if $x=\operatorname{cs}$ or $\operatorname{sc}$; $k^{\prime}\leq x\leq 1$ if $x=\operatorname{dn}$; $1\leq x\leq 1/k^{\prime}$ if $x=\operatorname{nd}$; $x^{2}\geq{k^{\prime}}^{2}$ if $x=\operatorname{ds}$; $0\leq x^{2}\leq 1/{k^{\prime}}^{2}$ if $x=\operatorname{sd}$. …