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11: 19.38 Approximations
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►Minimax polynomial approximations (§3.11(i)) for and in terms of with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸.
Approximations of the same type for and for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸.
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12: 29.7 Asymptotic Expansions
13: 3.4 Differentiation
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►For additional formulas involving values of and on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546).
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14: 26.2 Basic Definitions
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►If, for example, a permutation of the integers 1 through 6 is denoted by , then the cycles are , , and .
…The function also interchanges 3 and 6, and sends 4 to itself.
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►As an example, , , is a partition of .
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►As an example, is a partition of 13.
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►The example has six parts, three of which equal 1.
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15: 27.15 Chinese Remainder Theorem
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►Choose four relatively prime moduli , and of five digits each, for example , , , and .
…By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod ), (mod ), (mod ), and (mod ), respectively.
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 , , , and , where each has no more than five digits.
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16: 28.15 Expansions for Small
17: 20.7 Identities
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20.7.5
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20.7.9
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►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices .
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►See also Carlson (2011, §§1 and 4).
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20.7.24
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18: 12.4 Power-Series Expansions
19: 23 Weierstrass Elliptic and Modular
Functions
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