induced modulus

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… ►A divisor

d

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

). …

… ►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 …

… ►Let

{k^{\prime}}^{2}=1-k^{2}

and

c={\csc}^{2}\phi

with

0\leq\Re\phi\leq\ifrac{\pi}{2}

. … ►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}

. … ►where we assume

0\leq x^{2}\leq 1

x=\operatorname{sn}

\operatorname{cn}

, or

\operatorname{cd}

;

x^{2}\geq 1

x=\operatorname{ns}

\operatorname{nc}

, or

\operatorname{dc}

;

x

real if

x=\operatorname{cs}

\operatorname{sc}

;

k^{\prime}\leq x\leq 1

x=\operatorname{dn}

;

1\leq x\leq 1/k^{\prime}

x=\operatorname{nd}

;

x^{2}\geq{k^{\prime}}^{2}

x=\operatorname{ds}

;

0\leq x^{2}\leq 1/{k^{\prime}}^{2}

x=\operatorname{sd}

. …