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1: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 3.1 Arithmetics and Error Measures
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►The set of machine numbers
is the union of and the set
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►The lower and upper bounds for the absolute values of the nonzero machine numbers are given by
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Rounding
… ►Symmetric rounding or rounding to nearest of gives or , whichever is nearer to , with maximum relative error equal to the machine precision . …3: 27.15 Chinese Remainder Theorem
§27.15 Chinese Remainder Theorem
… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits. ►Even though the lengthy calculation is repeated four times, once for each modulus, most of it only uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory. Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …4: 27.16 Cryptography
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►Applications to cryptography rely on the disparity in computer time required to find large primes and to factor large integers.
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►With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer.
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►To decode, we must recover from .
…In other words, to recover from we simply raise
to the power and reduce modulo .
If and are known, and can be determined (mod ) by straightforward calculations that require only a few minutes of machine time.
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5: DLMF Project News
error generating summary6: 26.21 Tables
§26.21 Tables
►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions and partitions into distinct parts for up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to . ►Goldberg et al. (1976) contains tables of binomial coefficients to and Stirling numbers to .7: 4.13 Lambert -Function
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►For the definition of Stirling cycle numbers of the first kind see (26.13.3).
As
…For large enough the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side.
…As
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►For a generalization of the Lambert -function connected to the three-body problem see Scott et al. (2006), Scott et al. (2013) and Scott et al. (2014).
8: 9.19 Approximations
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►The constants and are chosen numerically, with a view to equalizing the effort required for summing the series.
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Martín et al. (1992) provides two simple formulas for approximating to graphical accuracy, one for , the other for .
Razaz and Schonfelder (1980) covers , , , . The Chebyshev coefficients are given to 30D.
Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of , stored at the nodes. and are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of , at the node. Similarly for , .
MacLeod (1994) supplies Chebyshev-series expansions to cover for and for . The Chebyshev coefficients are given to 20D.