moduli

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1: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{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}\pmod{m_{1}}

a_{2}\pmod{m_{2}}

a_{3}\pmod{m_{3}}

, and

a_{4}\pmod{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

(-\infty,0)

and

(1,\infty)

, 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: 4.21 Identities

… ►

§4.21(iv) Real and Imaginary Parts; Moduli

…

4: 4.35 Identities

… ►

§4.35(iv) Real and Imaginary Parts; Moduli

…

5: 29.1 Special Notation

… ► ►►►

$m, n, p$	nonnegative integers.
…
$K$ , ${K^{\prime}}$	complete elliptic integrals of the first kind with moduli $k,k^{\prime}$ , respectively (see §19.2(ii)).

…

6: 29.2 Differential Equations

… ►This equation has regular singularities at the points

2pK+(2q+1)\mathrm{i}{K^{\prime}}

, where

p,q\in\mathbb{Z}

, and

K

{K^{\prime}}

are the complete elliptic integrals of the first kind with moduli

k

k^{\prime}(=(1-k^{2})^{1/2})

, respectively; see §19.2(ii). …

7: 2.11 Remainder Terms; Stokes Phenomenon

… ►We now compute the forward differences

\Delta^{j}

j=0,1,2,\dots

, of the moduli of the rounded values of the first 6 neglected terms: …