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1: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

B_{n}\left(x\right)

, 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

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 3.1 Arithmetics and Error Measures

… ► ►The set of machine numbers

\mathbb{R}_{{\rm fl}}

is the union of

0

and the set … ►The lower and upper bounds for the absolute values of the nonzero machine numbers are given by … ►

Rounding

… ►Symmetric rounding or rounding to nearest of

x

gives

x_{-}

x_{+}

, whichever is nearer to

x

, with maximum relative error equal to the machine precision

\frac{1}{2}\epsilon_{M}=2^{-p}

. …

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

\pmod{m}

, 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

… ►Applications to cryptography rely on the disparity in computer time required to find large primes and to factor large integers. … ►With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. … ►To decode, we must recover

x

from

y

. …In other words, to recover

x

from

y

we simply raise

y

to the power

s

and reduce modulo

n

. If

p

and

q

are known,

s

and

y^{s}

can be determined (mod

n

) by straightforward calculations that require only a few minutes of machine time. …

5: DLMF Project News

error generating summary

6: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

for

m

up to 50 and

n

up to 25; extends Table 26.4.1 to

n=10

; tabulates Stirling numbers of the first and second kinds,

s\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts

\not\equiv\pm 2\pmod{5}

, partitions into parts

\not\equiv\pm 1\pmod{5}

, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to

\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}

. ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

7: 4.13 Lambert $W$ -Function

… ►For the definition of Stirling cycle numbers of the first kind

\genfrac{[}{]}{0.0pt}{}{n}{k}

see (26.13.3). As

\left|z\right|\to\infty

…For large enough

\left|z\right|

the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. …As

x\to 0-

… ►For a generalization of the Lambert

W

-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

… ►

•

Martín et al. (1992) provides two simple formulas for approximating $\operatorname{Ai}\left(x\right)$ to graphical accuracy, one for $-\infty<x\leq 0$ , the other for $0\leq x<\infty$ .

… ►The constants

a

and

b

are chosen numerically, with a view to equalizing the effort required for summing the series. … ►

•

Razaz and Schonfelder (1980) covers $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ . The Chebyshev coefficients are given to 30D.

… ►

•

Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ stored at the nodes. $\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ 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 $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ at the node. Similarly for $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ .

… ►

•

MacLeod (1994) supplies Chebyshev-series expansions to cover $\operatorname{Gi}\left(x\right)$ for $0\leq x<\infty$ and $\operatorname{Hi}\left(x\right)$ for $-\infty<x\leq 0$ . The Chebyshev coefficients are given to 20D.

9: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. … ►Apostol and Zuckerman (1951) uses congruences to construct magic squares. … ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

10: 27.20 Methods of Computation: Other Number-Theoretic Functions

§27.20 Methods of Computation: Other Number-Theoretic Functions

►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). … ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function

p\left(n\right)

for

n<N

. …To compute a particular value

p\left(n\right)

it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function

\tau\left(n\right)

, and the values can be checked by the congruence (27.14.20). …