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induced modulus

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1: 27.8 Dirichlet Characters
A divisor d of k is called an induced modulus for χ if … where χ 0 is a character (mod d ) for some induced modulus d for χ , and χ 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 χ ) , 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 2 , 1 / k 2 , - k 2 / k 2 , - k 2 / k 2 , and sin ϕ ( = ( z - x ) / z ) into k sin ϕ , - i tan ϕ , - i k tan ϕ , ( k sin ϕ ) / 1 - k 2 sin 2 ϕ , - i k sin ϕ / 1 - k 2 sin 2 ϕ . Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … Let r = 1 / x 2 . …
ϕ = arccos x / z = arcsin ( z - x ) / z ,
where we assume 0 x 2 1 if x = sn , cn , or cd ; x 2 1 if x = ns , nc , or dc ; x real if x = cs or sc ; k x 1 if x = dn ; 1 x 1 / k if x = nd ; x 2 k 2 if x = ds ; 0 x 2 1 / k 2 if x = sd . …