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1: 27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod m k ) , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod m ), where m is the product of the moduli. … Choose four relatively prime moduli m 1 , m 2 , m 3 , and m 4 of five digits each, for example 2 16 - 3 , 2 16 - 1 , 2 16 + 1 , and 2 16 + 3 . …Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers a 1 ( mod m 1 ) , a 2 ( mod m 2 ) , a 3 ( mod m 3 ) , and a 4 ( mod m 4 ) , where each a j has no more than five digits. …
2: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
§22.17(i) Real or Purely Imaginary Moduli
Jacobian elliptic functions with real moduli in the intervals ( - , 0 ) and ( 1 , ) , or with purely imaginary moduli are related to functions with moduli in the interval [ 0 , 1 ] by the following formulas. …
§22.17(ii) Complex Moduli
3: 4.35 Identities
§4.35(iv) Real and Imaginary Parts; Moduli
4: 4.21 Identities
§4.21(iv) Real and Imaginary Parts; Moduli
5: 29.1 Special Notation
m , n , p

nonnegative integers.

K , K

complete elliptic integrals of the first kind with moduli k , k , respectively (see §19.2(ii)).

6: 29.2 Differential Equations
This equation has regular singularities at the points 2 p K + ( 2 q + 1 ) i K , where p , q , and K , K are the complete elliptic integrals of the first kind with moduli k , k ( = ( 1 - k 2 ) 1 / 2 ) , respectively; see §19.2(ii). …
7: 2.11 Remainder Terms; Stokes Phenomenon
We now compute the forward differences Δ j , j = 0 , 1 , 2 , , of the moduli of the rounded values of the first 6 neglected terms: …